Devices operating on the principle of electromagnetic induction. Practical application of the phenomenon of electromagnetic induction

We already know that electricity, moving along the conductor, creates a magnetic field around it. On the basis of this phenomenon, man has invented and widely uses a wide variety of electromagnets. But the question arises: if electric charges, moving, cause the appearance magnetic field, but doesn't it work and vice versa?

That is, can a magnetic field cause an electric current to flow in a conductor? In 1831, Michael Faraday established that an electric current is generated in a closed conducting electrical circuit when a magnetic field changes. Such a current was called an induction current, and the phenomenon of the appearance of a current in a closed conducting circuit with a change in the magnetic field penetrating this circuit is called electromagnetic induction.

The phenomenon of electromagnetic induction

The name "electromagnetic" itself consists of two parts: "electro" and "magnetic". Electrical and magnetic phenomena are inextricably linked with each other. And if the electric charges, moving, change the magnetic field around them, then the magnetic field, changing, willy-nilly make the electric charges move, forming an electric current.

In this case, it is the changing magnetic field that causes the occurrence of an electric current. A permanent magnetic field will not cause movement electric charges, and accordingly, the induction current is not formed. More detailed consideration phenomena of electromagnetic induction, the derivation of formulas and the law of electromagnetic induction refers to the course of the ninth grade.

Application of electromagnetic induction

In this article, we will talk about the use of electromagnetic induction. The operation of many motors and current generators is based on the use of the laws of electromagnetic induction. The principle of their work is quite simple to understand.

A change in the magnetic field can be caused, for example, by moving a magnet. Therefore, if a magnet is moved inside a closed circuit by some third-party influence, then a current will appear in this circuit. So you can create a current generator.

If, on the contrary, a current from a third-party source is passed through the circuit, then the magnet inside the circuit will begin to move under the influence of the magnetic field generated by the electric current. In this way, an electric motor can be assembled.

The current generators described above convert mechanical energy into electrical energy at power plants. Mechanical energy is the energy of coal, diesel fuel, wind, water and so on. Electricity is supplied by wires to consumers and there it is converted back into mechanical energy in electric motors.

The electric motors of vacuum cleaners, hair dryers, mixers, coolers, electric meat grinders and numerous other devices that we use daily are based on the use of electromagnetic induction and magnetic forces. There is no need to talk about the use of these same phenomena in industry, it is clear that it is ubiquitous.

Khudoley Andrey, Khnykov Igor

Practical application of the phenomenon of electromagnetic induction.

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Electromagnetic induction in modern technology Performed by students of 11 "A" class MOUSOSH No. 2 of the city of Suvorov Khnykov Igor, Khudoley Andrey

The phenomenon of electromagnetic induction was discovered on August 29, 1831 by Michael Faraday. The phenomenon of electromagnetic induction consists in the occurrence of an electric current in a conducting circuit, which either rests in a magnetic field that changes in time, or moves in a constant magnetic field in such a way that the number of magnetic induction lines penetrating the circuit changes.

The EMF of electromagnetic induction in a closed circuit is numerically equal and opposite in sign to the rate of change of the magnetic flux through the surface bounded by this circuit. Direction induction current(as well as the value of the EMF), is considered positive if it coincides with the selected direction of bypassing the circuit.

Faraday's experiment A permanent magnet is inserted into or removed from a coil connected to a galvanometer. When the magnet moves in the circuit, an electric current arises. Within one month, Faraday experimentally discovered all the essential features of the phenomenon of electromagnetic induction. At present, Faraday's experiments can be carried out by anyone.

The main sources of the electromagnetic field The main sources of the electromagnetic field are: Power lines. Wiring (inside buildings and structures). Household electrical appliances. Personal computers. TV and radio transmitting stations. Satellite and cellular communications (devices, repeaters). Electric transport. radar installations.

Power lines The wires of an operating power line create an electromagnetic field of industrial frequency (50 Hz) in the adjacent space (at distances of the order of tens of meters from the wire). Moreover, the field strength near the line can vary over a wide range, depending on its electrical load. Actually borders sanitary protection zone are installed along the boundary line furthest from the wires with a maximum electric field strength of 1 kV / m.

Electrical wiring Electrical wiring includes: power cables for building life support systems, power distribution wires, as well as branching boards, power boxes and transformers. Electrical wiring is the main source of the industrial frequency electromagnetic field in residential premises. In this case, the level of the electric field strength emitted by the source is often relatively low (does not exceed 500 V/m).

Household appliances Sources of electromagnetic fields are all Appliances operating using electric current. At the same time, the level of radiation varies over the widest range, depending on the model, the device device and the specific mode of operation. Also, the level of radiation strongly depends on the power consumption of the device - the higher the power, the higher the level of the electromagnetic field during the operation of the device. The electric field strength near household appliances does not exceed tens of V/m.

Personal Computers The primary source of adverse health effects for a computer user is the monitor's display device (VOD). In addition to the monitor and the system unit, a personal computer may also include a large number of other devices (such as printers, scanners, surge protectors, etc.). All these devices work with the use of electric current, which means that they are sources of an electromagnetic field.

The electromagnetic field of personal computers has the most complex wave and spectral composition and is difficult to measure and quantify. It has magnetic, electrostatic and radiation components (in particular, the electrostatic potential of a person sitting in front of a monitor can range from -3 to +5 V). Considering the condition that personal computers now widely used in all industries human activity, their impact on human health is subject to careful study and control

Television and radio transmitting stations A significant number of radio broadcasting stations and centers of various affiliations are currently located on the territory of Russia. Transmitting stations and centers are located in areas specially designated for them and can occupy quite large territories(up to 1000 ha). By their structure, they include one or more technical buildings, where radio transmitters are located, and antenna fields, on which up to several dozen antenna-feeder systems (AFS) are located. Each system includes a radiating antenna and a feeder line that brings the broadcast signal.

Satellite communication Satellite communication systems consist of a transmitting station on the Earth and satellites - repeaters in orbit. Transmitting satellite communication stations emit a narrowly directed wave beam, the energy flux density in which reaches hundreds of W/m. Satellite communication systems create high electromagnetic field strengths at considerable distances from antennas. For example, a station with a power of 225 kW, operating at a frequency of 2.38 GHz, creates an energy flux density of 2.8 W/m2 at a distance of 100 km. The scattering of energy relative to the main beam is very small and occurs most of all in the area of ​​\u200b\u200bthe direct placement of the antenna.

Cellular communication Cellular radiotelephony is today one of the most intensively developing telecommunication systems. The main elements of the system cellular communication are base stations and mobile radiotelephones. Base stations maintain radio communication with mobile devices, as a result of which they are sources of an electromagnetic field. The system uses the principle of dividing the coverage area into zones, or so-called "cells", with a radius of km.

The radiation intensity of the base station is determined by the load, that is, the presence of owners cell phones in the service area of ​​a particular base station and their desire to use the phone for a conversation, which, in turn, depends fundamentally on the time of day, the location of the station, the day of the week and other factors. At night, the loading of stations is almost zero. The radiation intensity of mobile devices depends largely on the state of the communication channel "mobile radiotelephone - base station" (the greater the distance from the base station, the higher the radiation intensity of the device).

Electric transport Electric transport (trolleybuses, trams, subway trains, etc.) is a powerful source of electromagnetic field in the Hz frequency range. At the same time, in the vast majority of cases, the traction electric motor acts as the main emitter (for trolleybuses and trams, air current collectors compete with the electric motor in terms of the strength of the radiated electric field).

Radar installations Radar and radar installations usually have reflector-type antennas (“dishes”) and emit a narrowly directed radio beam. Periodic movement of the antenna in space leads to spatial discontinuity of radiation. There is also a temporary discontinuity of radiation due to the cyclic operation of the radar for radiation. They operate at frequencies from 500 MHz to 15 GHz, but some special installations can operate at frequencies up to 100 GHz or more. Due to the special nature of the radiation, they can create zones with a high energy flux density (100 W/m2 or more) on the ground.

Metal detectors Technologically, the principle of operation of a metal detector is based on the phenomenon of registering an electromagnetic field that is created around any metal object when it is placed in an electromagnetic field. This secondary electromagnetic field differs both in intensity (field strength) and in other parameters. These parameters depend on the size of the object and its conductivity (gold and silver have much better conductivity than, for example, lead) and, of course, on the distance between the metal detector antenna and the object itself (depth of occurrence).

The above technology determined the composition of the metal detector: it consists of four main blocks: an antenna (sometimes the emitting and receiving antennas are different, and sometimes they are the same antenna), an electronic processing unit, an information output unit (visual - LCD display or arrow indicator and audio - speaker or headphone jack) and power supply.

Metal detectors are: Search Inspection For construction purposes

Search This metal detector is designed to search for all kinds of metal objects. As a rule, these are the largest in size, cost and, of course, in terms of the functions of the model. This is due to the fact that sometimes you need to find objects at a depth of up to several meters in the thickness of the earth. The powerful antenna is able to create a high level of electromagnetic field and detect even the slightest currents at great depths with high sensitivity. For example, a search metal detector detects a metal coin at a depth of 2-3 meters in the earth, which may even contain ferruginous geological compounds.

Inspection It is used by special services, customs officers and security officers of various organizations to search for metal objects (weapons, precious metals, wires of explosive devices, etc.) hidden on the body and in clothes of a person. These metal detectors are distinguished by compactness, ease of use, the presence of modes such as silent vibration of the handle (so that the searched person does not know that the search officer has found something). The range (depth) of detection of a ruble coin in such metal detectors reaches 10-15 cm.

Also wide use received arched metal detectors that look like an arch and require a person to pass through it. Along them vertical walls ultra-sensitive antennas have been laid that detect metal objects at all levels of human growth. They are usually installed in front of places of cultural entertainment, in banks, institutions, etc. main feature arched metal detectors - high sensitivity (adjustable) and high speed of processing the flow of people.

For building purposes This class metal detectors with the help of sound and light alarms helps builders find metal pipes, structural or drive elements located both in the thickness of the walls and behind partitions and false panels. Some metal detectors for construction purposes are often combined in one device with detectors wooden construction, voltage detectors on current-carrying wires, leakage detectors, etc.

Broadcasting. An alternating magnetic field, excited by a changing current, creates in the surrounding space electric field, which in turn excites a magnetic field, and so on. Mutually generating each other, these fields form a single variable electromagnetic field - 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 occupied by radio waves, light, x-rays other electromagnetic radiation. They are usually characterized by continuously interconnected electric and magnetic fields.

Synchrophasotrons.Currently, 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.

Flowmeters - counters. 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 stator poles there is a constant magnetic flux piercing anchor. 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 rule " right hand". In this case, a positive potential arises on one brush relative to the second. If a load is connected to the generator terminals, then a current will flow in it.

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

TRANSFORMERS.) - static electromagnetic device having two or more inductively coupled windings and intended to be converted by electromagnetic induction of one or more alternating current systems into one or more other alternating current systems.

The occurrence of induction current in a rotating circuit and its application.

The phenomenon of electromagnetic induction is used to convert mechanical energy into electrical energy. For this purpose, are used generators, operating principle

which can be considered on the example of a flat frame rotating in a uniform magnetic field

Let the frame rotate in a uniform magnetic field (B = const) uniformly with angular velocity u = const.

Magnetic flux coupled to a frame area S, at any point in time t equals

where a - ut- the angle of rotation of the frame at the time t(the origin is chosen so that at /. = 0 there is a = 0).

When the frame rotates, a variable induction emf will appear in it

changing with time according to the harmonic law. EMF %" maximum at sin Wt= 1, i.e.

Thus, if in a homogeneous

If the frame rotates uniformly in a magnetic field, then a variable EMF arises in it, which changes according to the harmonic law.

The process of converting mechanical energy into electrical energy is reversible. If a current is passed through a frame placed in a magnetic field, a torque will act on it and the frame will begin to rotate. This principle is based on the operation of electric motors designed to convert electrical energy into mechanical.

Ticket 5.

Magnetic field in matter.

Experimental studies showed that all substances to a greater or lesser extent have magnetic properties. If two turns with currents are placed in any medium, then the strength of the magnetic interaction between the currents changes. This experience shows that the induction of the magnetic field created by electric currents in a substance differs from the induction of the magnetic field created by the same currents in a vacuum.

The physical quantity showing how many times the magnetic field induction in a homogeneous medium differs in absolute value from the magnetic field induction in vacuum is called magnetic permeability:

The magnetic properties of substances are determined by the magnetic properties of atoms or elementary particles(electrons, protons and neutrons) that make up atoms. It is currently established that magnetic properties protons and neutrons are almost 1000 times weaker than the magnetic properties of electrons. Therefore, the magnetic properties of substances are mainly determined by the electrons that make up the atoms.

Substances are extremely diverse in their magnetic properties. In most substances, these properties are weakly expressed. Weakly magnetic substances are divided into two large groups - paramagnets and diamagnets. They differ in that when introduced into an external magnetic field, paramagnetic samples are magnetized so that their own magnetic field turns out to be directed along the external field, and diamagnetic samples are magnetized against the external field. Therefore, for paramagnets μ > 1, and for diamagnets μ< 1. Отличие μ от единицы у пара- и диамагнетиков чрезвычайно мало. Например, у алюминия, который относится к парамагнетикам, μ – 1 ≈ 2,1·10–5, у хлористого железа (FeCl3) μ – 1 ≈ 2,5·10–3. К парамагнетикам относятся также платина, воздух и многие другие вещества. К диамагнетикам относятся медь (μ – 1 ≈ –3·10–6), вода (μ – 1 ≈ –9·10–6), висмут (μ – 1 ≈ –1,7·10–3) и другие вещества. Образцы из пара- и диамагнетика, помещенные в неоднородное магнитное поле между полюсами электромагнита, ведут себя по-разному – парамагнетики втягиваются в область сильного поля, диамагнетики – выталкиваются (рис. 1.19.1).

Problems of magnetostatics in matter.

Magnetic characteristics of matter - magnetization vector, magnetic

susceptibility and magnetic permeability of a substance.

Magnetization vector - the magnetic moment of an elementary volume used to describe the magnetic state of matter. In relation to the direction of the magnetic field vector, longitudinal magnetization and transverse magnetization are distinguished. The transverse magnetization reaches significant values ​​in anisotropic magnets, and is close to zero in isotropic magnets. Therefore, in the latter it is possible to express the magnetization vector in terms of the magnetic field strength and the coefficient x called the magnetic susceptibility:

Magnetic susceptibility - physical quantity characterizing the relationship between the magnetic moment (magnetization) of a substance and the magnetic field in this substance.

Magnetic permeability - a physical quantity that characterizes the relationship between magnetic induction and magnetic field strength in a substance.

Usually denoted Greek letter. It can be either a scalar (for isotropic substances) or a tensor (for anisotropic substances).

AT general view is injected as a tensor like this:

Ticket 6.

Classification of magnets

magnets substances are called that are capable of acquiring their own magnetic field in an external magnetic field, i.e., being magnetized. The magnetic properties of matter are determined by the magnetic properties of electrons and atoms (molecules) of matter. According to their magnetic properties, magnets are divided into three main groups: diamagnets, paramagnets and ferromagnets.

1. Magnetics with linear dependence:

1) Paramagnets - substances that are weakly magnetized in a magnetic field, and the resulting field in paramagnets is stronger than in vacuum, the magnetic permeability of paramagnets m\u003e 1; Such properties are possessed by aluminum, platinum, oxygen, etc.;

paramagnets ,

2) Diamagnets - substances that are weakly magnetized against the field, that is, the field in diamagnets is weaker than in vacuum, the magnetic permeability m< 1. К диамагнетикам относятся медь, серебро, висмут и др.;

diamagnets ;

With non-linear dependence:

3) ferromagnets - substances that can be strongly magnetized in a magnetic field,. These are iron, cobalt, nickel and some alloys. 2.

Ferromagnets.

Depends on background and is a function of tension; exist hysteresis.

And it can reach high values ​​in comparison with para- and diamagnets.

The total current law for a magnetic field in matter (theorem of the circulation of the vector B)

Where I and I "are, respectively, the algebraic sums of macrocurrents (conduction currents) and microcurrents (molecular currents) covered by an arbitrary closed loop L. Thus, the circulation of the magnetic induction vector B along an arbitrary closed loop is equal to the algebraic sum of conduction currents and molecular currents covered by this The vector B thus characterizes the resulting field created by both macroscopic currents in conductors (conduction currents) and microscopic currents in magnets, so the lines of the magnetic induction vector B have no sources and are closed.

Magnetic field intensity vector and its circulation.

The magnetic field strength - (standard designation H) is a vector physical quantity equal to the difference between the magnetic induction vector B and the magnetization vector M.

In SI: where is the magnetic constant

Conditions at the interface between two media

Exploring the relationship between vectors E and D at the interface between two homogeneous isotropic dielectrics (whose permittivities are ε 1 and ε 2) in the absence of free charges on the boundary.

Replacing the projections of the vector E vector projections D, divided by ε 0 ε, we get

construct a straight cylinder of negligible height at the interface between two dielectrics (Fig. 2); one base of the cylinder is in the first dielectric, the other is in the second. The bases of ΔS are so small that within each of them the vector D the same. According to the Gauss theorem for electrostatic field in dielectric

(normal n and n" opposite to the bases of the cylinder). So

Replacing the projections of the vector D vector projections E, multiplied by ε 0 ε, we obtain

Hence, when passing through the interface between two dielectric media, the tangential component of the vector E(Е τ) and the normal component of the vector D(D n) change continuously (do not experience a jump), and the normal component of the vector E(E n) and the tangential component of the vector D(D τ) experience a jump.

From conditions (1) - (4) for the constituent vectors E and D we see that the lines of these vectors experience a break (refract). Let's find how the angles α 1 and α 2 are related (in Fig. 3 α 1 > α 2). Using (1) and (4), Е τ2 = Е τ1 and ε 2 E n2 = ε 1 E n1 . Let's decompose the vectors E 1 and E 2 into tangential and normal components at the interface. From fig. 3 we see that

Taking into account the conditions written above, we find the law of refraction of tension lines E(and hence the displacement lines D)

From this formula, we can conclude that, entering a dielectric with a higher permittivity, the lines E and D move away from the normal.

Ticket 7.

Magnetic moments of atoms and molecules.

Elementary particles have a magnetic moment, atomic nuclei, electron shells of atoms and molecules. The magnetic moment of elementary particles (electrons, protons, neutrons and others), as shown by quantum mechanics, is due to the existence of their own mechanical moment - spin. The magnetic moment of the nuclei is made up of their own (spin) magnetic moment of the protons and neutrons that form these nuclei, as well as the magnetic moment associated with their orbital motion inside the nucleus. Magnetic moment electron shells atoms and molecules are made up of spin and orbital magnetic moment of electrons. The spin magnetic moment of an electron msp can have two equal and oppositely directed projections on the direction of the external magnetic field H. The absolute value of the projection

where mb = (9.274096 ±0.000065) 10-21erg/gs - Boron magneton where h - Planck's constant, e and me - the charge and mass of the electron, c - the speed of light; SH is the projection of the spin mechanical moment on the direction of the field H. The absolute value of the spin magnetic moment

types of magnets.

MAGNETIC, a substance with magnetic properties, which are determined by the presence of its own or induced by an external magnetic field magnetic moments, as well as the nature of the interaction between them. There are diamagnets, in which the external magnetic field creates a resulting magnetic moment directed opposite to the external field, and paramagnets, in which these directions coincide.

Diamagnets- substances that are magnetized against the direction of an external magnetic field. In the absence of an external magnetic field, diamagnets are non-magnetic. Under the action of an external magnetic field, each atom of a diamagnet acquires a magnetic moment I (and each mole of a substance acquires a total magnetic moment), proportional to the magnetic induction H and directed towards the field.

Paramagnets- substances that are magnetized in an external magnetic field in the direction of the external magnetic field. Paramagnets are weakly magnetic substances, the magnetic permeability differs slightly from unity.

Atoms (molecules or ions) of a paramagnet have their own magnetic moments, which, under the action of external fields, are oriented along the field and thereby create a resulting field that exceeds the external one. Paramagnets are drawn into a magnetic field. In the absence of an external magnetic field, a paramagnet is not magnetized, since due to thermal motion, the intrinsic magnetic moments of atoms are oriented completely randomly.

Orbital magnetic and mechanical moments.

An electron in an atom moves around the nucleus. In classical physics, the movement of a point along a circle corresponds to the angular momentum L=mvr, where m is the mass of the particle, v is its velocity, r is the radius of the trajectory. AT quantum mechanics this formula is inapplicable, since the radius and speed are both uncertain (see "Uncertainty relation"). But the magnitude of the angular momentum itself exists. How to define it? It follows from the quantum mechanical theory of the hydrogen atom that the modulus of the angular momentum of an electron can take the following discrete values:

where l is the so-called orbital quantum number, l = 0, 1, 2, … n-1. Thus, the angular momentum of an electron, like energy, is quantized, i.e. takes discrete values. Note that for large values quantum number l (l >>1) equation (40) will take the form . This is nothing but one of N. Bohr's postulates.

From the quantum mechanical theory of the hydrogen atom follows another important conclusion: the projection of the angular momentum of the electron onto some given direction in space z (for example, the direction lines of force magnetic or electric field) is also quantized according to the rule:

where m = 0, ± 1, ± 2, …± l is the so-called magnetic quantum number.

An electron moving around the nucleus is an elementary circular electric current. This current corresponds to the magnetic moment pm. Obviously, it is proportional to the mechanical angular momentum L. The ratio of the magnetic moment pm of an electron to the mechanical angular momentum L is called the gyromagnetic ratio. For an electron in a hydrogen atom

the minus sign indicates that the vectors of the magnetic and mechanical moments are directed in opposite directions). From here you can find the so-called orbital magnetic moment of the electron:

hydromagnetic relationship.

Ticket 8.

Atom in an external magnetic field. Precession of the plane of the orbit of an electron in an atom.

When an atom is introduced into a magnetic field with induction, an electron moving in an orbit equivalent to a closed circuit with current is affected by a moment of forces:

The vector of the orbital magnetic moment of the electron changes similarly:

,

(6.2.3)

It follows from this that the vectors and , and the orbit itself precesses around the direction of the vector . Figure 6.2 shows the precessional motion of the electron and its orbital magnetic moment, as well as the additional (precessional) motion of the electron.

This precession is called Larmor precession . The angular velocity of this precession depends only on the magnetic field induction and coincides with it in direction.

,

(6.2.4)

Induced orbital magnetic moment.

Larmor's theorem:the only result of the influence of a magnetic field on the orbit of an electron in an atom is the precession of the orbit and the vector - the orbital magnetic moment of the electron with an angular velocity around the axis passing through the nucleus of the atom parallel to the magnetic field induction vector.

The precession of the orbit of an electron in an atom leads to the appearance of an additional orbital current directed opposite to the current I:

where is the area of ​​the projection of the electron orbit onto the plane perpendicular to the vector . The minus sign says that it is opposite to the vector. Then the total orbital momentum of the atom is:

,

diamagnetic effect.

The diamagnetic effect is an effect in which the components of the magnetic fields of atoms add up and form their own magnetic field of the substance, which weakens the external magnetic field.

Since the diamagnetic effect is due to the action of an external magnetic field on the electrons of the atoms of a substance, diamagnetism is characteristic of all substances.

The diamagnetic effect occurs in all substances, but if the molecules of the substance have their own magnetic moments, which are oriented in the direction of the external magnetic field and enhance it, then the diamagnetic effect is blocked by a stronger paramagnetic effect and the substance turns out to be a paramagnet.

The diamagnetic effect occurs in all substances, but if the molecules of the substance have their own magnetic moments, which are oriented in the direction of the external magnetic field and increase erOj, then the diamagnetic effect is overlapped by a stronger paramagnetic effect and the substance turns out to be a paramagnet.

Larmor's theorem.

If an atom is placed in an external magnetic field with induction (Fig. 12.1), then the electron moving in orbit will be affected by the rotational moment of forces, seeking to establish the magnetic moment of the electron in the direction of the magnetic field lines (mechanical moment - against the field).

Ticket 9

9.Strongly magnetic substances - ferromagnets- substances with spontaneous magnetization, i.e. they are magnetized even in the absence of an external magnetic field. In addition to their main representative, iron, ferromagnets include, for example, cobalt, nickel, gadolinium, their alloys and compounds.

For ferromagnets, the dependence J from H pretty complicated. As you rise H magnetization J first grows rapidly, then more slowly, and finally, the so-called magnetic saturationJ us, no longer dependent on the strength of the field.

Magnetic induction AT=m 0 ( H+J) in weak fields grows rapidly with increasing H due to increased J, but in strong fields, since the second term is constant ( J=J us), AT grows with the increase H according to a linear law.

An essential feature of ferromagnets is not only large values ​​of m (for example, for iron - 5000), but also the dependence of m on H. Initially, m grows with increasing H, then, reaching a maximum, it begins to decrease, tending to 1 in the case of strong fields (m= B/(m 0 H)= 1+J/N, so when J=J us =const with growth H attitude J/H->0, and m.->1).

Feature ferromagnets also consists in the fact that for them the dependence J from H(and consequently, and B from H) is determined by the prehistory of the magnetization of the ferromagnet. This phenomenon has been named magnetic hysteresis. If you magnetize a ferromagnet to saturation (point 1 , rice. 195) and then start to reduce the tension H magnetizing field, then, as experience shows, a decrease J described by a curve 1 -2, above the curve 1 -0. At H=0 J different from zero, i.e. observed in a ferromagnet residual magnetizationJoc. The presence of residual magnetization is associated with the existence permanent magnets. The magnetization vanishes under the action of the field H C , having a direction opposite to the field that caused the magnetization.

tension H C called coercive force.

With a further increase in the opposite field, the ferromagnet is remagnetized (curve 3-4), and at H=-H we reach saturation (point 4). Then the ferromagnet can be demagnetized again (curve 4-5 -6) and remagnetize to saturation (curve 6- 1 ).

Thus, under the action of an alternating magnetic field on a ferromagnet, the magnetization J changes in accordance with the curve 1 -2-3-4-5-6-1, which is called hysteresis loop. Hysteresis leads to the fact that the magnetization of a ferromagnet is not a single-valued function of H, i.e., the same value H matches multiple values J.

Different ferromagnets give different hysteresis loops. ferromagnets with low (ranging from a few thousandths to 1-2 A/cm) coercive force H C(with a narrow hysteresis loop) are called soft, with a large (from several tens to several thousand amperes per centimeter) coercive force (with a wide hysteresis loop) - tough. Quantities H C, J oc and m max determine the applicability of ferromagnets for various practical purposes. So, hard ferromagnets (for example, carbon and tungsten steels) are used to make permanent magnets, and soft ones (for example, soft iron, iron-nickel alloy) are used to make transformer cores.

Ferromagnets have another essential feature: for each ferromagnet there is a certain temperature, called Curie point, at which it loses its magnetic properties. When the sample is heated above the Curie point, the ferromagnet transforms into an ordinary paramagnet.

The process of magnetization of ferromagnets is accompanied by a change in its linear dimensions and volume. This phenomenon has been named magnetostriction.

The nature of ferromagnetism. According to the ideas of Weiss, ferromagnets at temperatures below the Curie point have spontaneous magnetization, regardless of the presence of an external magnetizing field. Spontaneous magnetization, however, is in apparent contradiction with the fact that many ferromagnetic materials, even at temperatures below the Curie point, are not magnetized. To eliminate this contradiction, Weiss introduced the hypothesis that a ferromagnet below the Curie point is divided into big number small macroscopic areas - domains, spontaneously magnetized to saturation.

In the absence of an external magnetic field, the magnetic moments of individual domains are randomly oriented and compensate each other, so the resulting magnetic moment of a ferromagnet zero and a ferromagnet is not magnetized. An external magnetic field orients along the field the magnetic moments not of individual atoms, as is the case in the case of paramagnets, but of entire regions of spontaneous magnetization. Therefore, with the growth H magnetization J and magnetic induction AT already in rather weak fields grow very rapidly. This also explains the increase in m ferromagnets to a maximum value in weak fields. Experiments have shown that the dependence of B on R is not as smooth as shown in Fig. 193, but has a stepped view. This indicates that inside the ferromagnet, the domains turn in a jump across the field.

When the external magnetic field is weakened to zero, ferromagnets retain residual magnetization, since thermal motion is not able to quickly disorient the magnetic moments of such large formations as domains. Therefore, the phenomenon of magnetic hysteresis is observed (Fig. 195). In order to demagnetize a ferromagnet, a coercive force must be applied; shaking and heating of the ferromagnet also contribute to demagnetization. The Curie point turns out to be the temperature above which the destruction of the domain structure occurs.

The existence of domains in ferromagnets has been proven experimentally. A direct experimental method for their observation is powder figure method. An aqueous suspension of a fine ferromagnetic powder (for example, magnetite) is applied to the carefully polished surface of a ferromagnet. Particles settle mainly in places of maximum inhomogeneity of the magnetic field, i.e., at the boundaries between domains. Therefore, the settled powder outlines the boundaries of the domains, and a similar picture can be photographed under a microscope. Linear dimensions domains were equal to 10 -4 -10 -2 cm.

The principle of operation of transformers, used to increase or decrease the voltage of alternating current, is based on the phenomenon of mutual induction.

Primary and secondary coils (windings), having respectively n 1 and N 2 turns, mounted on a closed iron core. Since the ends of the primary winding are connected to an alternating voltage source with emf. ξ 1 , then it arises alternating current I 1 , creating an alternating magnetic flux F in the transformer core, which is almost completely localized in the iron core and, therefore, almost completely penetrates the turns of the secondary winding. A change in this flux causes the emf to appear in the secondary winding. mutual induction, and in the primary - emf. self-induction.

Current I 1 primary winding is determined according to Ohm's law: where R 1 is the resistance of the primary winding. Voltage drop I 1 R 1 on resistance R 1 for rapidly changing fields is small compared to each of the two emfs, therefore . emf mutual induction that occurs in the secondary winding,

We get that emf, arising in the secondary winding, where the minus sign shows that the emf. in the primary and secondary windings are opposite in phase.

The ratio of the number of turns N 2 /N 1 , showing how many times the emf. more (or less) in the secondary winding of the transformer than in the primary is called transformation ratio.

Neglecting energy losses, which in modern transformers do not exceed 2% and are mainly associated with the release of Joule heat in the windings and the appearance of eddy currents, and applying the energy conservation law, we can write that the current powers in both transformer windings are almost the same: ξ 2 I 2 »ξ 1 I 1 , find ξ 2 /ξ 1 = I 1 /I 2 = N 2 /N 1, i.e., the currents in the windings are inversely proportional to the number of turns in these windings.

If a N 2 /N 1 >1, then we are dealing with step up transformer, increasing the emf variable. and lowering current (used, for example, for the transmission of electricity over long distances, since in this case the losses for Joule heat, proportional to the square of the current strength, are reduced); if N 2 /N 1 <1, then we are dealing with step down transformer, reducing emf. and increasing current (used, for example, in electric welding, since it requires a large current at low voltage).

A transformer with one winding is called autotransformer. In the case of a step-up autotransformer, the e.m.f. is supplied to a part of the winding, and the secondary emf. removed from the entire winding. In a step-down autotransformer, the mains voltage is applied to the entire winding, and the secondary emf. removed from the winding.

11. Harmonic fluctuation - the phenomenon of a periodic change in a quantity, in which the dependence on the argument has the character of a sine or cosine function. For example, a quantity that varies in time as follows harmonically fluctuates:

Or, where x is the value of the changing quantity, t is time, the remaining parameters are constant: A is the amplitude of the oscillations, ω is the cyclic frequency of the oscillations, is the full phase of the oscillations, is the initial phase of the oscillations. Generalized harmonic oscillation in differential form

Types of vibrations:

Free oscillations are performed under the action of the internal forces of the system after the system has been taken out of equilibrium. For free oscillations to be harmonic, it is necessary that the oscillatory system be linear (described by linear equations of motion), and there should be no energy dissipation in it (the latter would cause damping).

Forced oscillations are performed under the influence of an external periodic force. For them to be harmonic, it is sufficient that the oscillatory system be linear (described by linear equations of motion), and the external force itself changes over time as a harmonic oscillation (that is, that the time dependence of this force is sinusoidal).

Mechanical harmonic oscillation is a rectilinear non-uniform movement in which the coordinates of an oscillating body (material point) change according to the cosine or sine law depending on time.

According to this definition, the law of coordinate change depending on time has the form:

where wt is the value under the cosine or sine sign; w is the coefficient, the physical meaning of which will be revealed below; A is the amplitude of mechanical harmonic oscillations. Equations (4.1) are the main kinematic equations of mechanical harmonic vibrations.

Periodic changes in the intensity E and induction B are called electromagnetic oscillations. Electromagnetic oscillations are radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, x-rays, gamma rays.

Formula derivation

Electromagnetic waves as a universal phenomenon were predicted by the classical laws of electricity and magnetism, known as Maxwell's equations. If you look closely at Maxwell's equation in the absence of sources (charges or currents), you will find that along with the possibility that nothing will happen, the theory also allows for non-trivial solutions for changing electric and magnetic fields. Let's start with Maxwell's equations for vacuum:

where is a vector differential operator (nabla)

One of the solutions is the simplest.

To find another, more interesting solution, we use the vector identity, which is valid for any vector, in the form:

To see how we can use it, let's take the swirl operation from expression (2):

The left side is equivalent to:

where we simplify using equation (1) above.

The right side is equivalent to:

Equations (6) and (7) are equal, so these results in a vector-valued differential equation for an electric field, namely

Applying similar initial results in a similar differential equation for a magnetic field:

These differential equations are equivalent to the wave equation:

where c0 is the speed of the wave in vacuum; f describes the displacement.

Or even simpler: where is the d'Alembert operator:

Note that in the case of electric and magnetic fields, the speed is:

The differential equation of harmonic oscillations of a material point , or , where m is the mass of the point; k - coefficient of quasi-elastic force (k=тω2).

The harmonic oscillator in quantum mechanics is a quantum analogue of a simple harmonic oscillator, while considering not the forces acting on the particle, but the Hamiltonian, that is, the total energy of the harmonic oscillator, and the potential energy is assumed to be quadratically dependent on the coordinates. Accounting for the following terms in the expansion of the potential energy with respect to the coordinate leads to the concept of an anharmonic oscillator

A harmonic oscillator (in classical mechanics) is a system that, when displaced from an equilibrium position, experiences a restoring force F proportional to the displacement x (according to Hooke's law):

where k is a positive constant describing the stiffness of the system.

The Hamiltonian of a quantum oscillator of mass m, whose natural frequency is ω, looks like this:

In coordinate representation , . The problem of finding the energy levels of a harmonic oscillator is reduced to finding such numbers E for which the following partial differential equation has a solution in the class of square-integrable functions.

An anharmonic oscillator is understood as an oscillator with a non-quadratic dependence of the potential energy on the coordinate. The simplest approximation of an anharmonic oscillator is the potential energy approximation up to the third term in the Taylor series:

12. Spring pendulum - a mechanical system consisting of a spring with a coefficient of elasticity (stiffness) k (Hooke's law), one end of which is rigidly fixed, and at the other there is a load of mass m.

When an elastic force acts on a massive body, returning it to the equilibrium position, it oscillates around this position. Such a body is called a spring pendulum. The vibrations are caused by an external force. Oscillations that continue after the external force has ceased to act are called free oscillations. Oscillations caused by the action of an external force are called forced. In this case, the force itself is called compelling.

In the simplest case, a spring pendulum is a rigid body moving along a horizontal plane, attached to a wall by a spring.

Newton's second law for such a system in the absence of external forces and friction forces has the form:

If the system is influenced by external forces, then the oscillation equation will be rewritten as follows:

Where f(x) is the resultant of external forces related to the unit mass of the load.

In the case of attenuation proportional to the speed of oscillations with a coefficient c:

Spring pendulum period:

A mathematical pendulum is an oscillator, which is a mechanical system consisting of a material point located on a weightless inextensible thread or on a weightless rod in a uniform field of gravitational forces. The period of small natural oscillations of a mathematical pendulum of length l, motionlessly suspended in a uniform gravitational field with free fall acceleration g, is equal to and does not depend on the amplitude and mass of the pendulum.

The differential equation of a spring pendulum x=Асos (wot+jo).

Pendulum equation

Oscillations of a mathematical pendulum are described by an ordinary differential equation of the form

where w is a positive constant determined solely from the parameters of the pendulum. unknown function; x(t) is the angle of deviation of the pendulum at the moment from the lower equilibrium position, expressed in radians; , where L is the suspension length, g is the free fall acceleration. The equation for small oscillations of the pendulum near the lower equilibrium position (the so-called harmonic equation) has the form:

A pendulum that makes small oscillations moves along a sinusoid. Since the equation of motion is an ordinary DE of the second order, to determine the law of motion of the pendulum, it is necessary to set two initial conditions - the coordinate and the velocity, from which two independent constants are determined:

where A is the amplitude of the pendulum oscillations, is the initial phase of the oscillations, w is the cyclic frequency, which is determined from the equation of motion. The movement of the pendulum is called harmonic oscillation.

A physical pendulum is an oscillator, which is a rigid body that oscillates in the field of any forces about a point that is not the center of mass of this body, or a fixed axis perpendicular to the direction of the forces and not passing through the center of mass of this body.

Moment of inertia about the axis passing through the suspension point:

Neglecting the resistance of the medium, the differential equation for the oscillations of a physical pendulum in the field of gravity is written as follows:

The reduced length is a conditional characteristic of a physical pendulum. It is numerically equal to the length of the mathematical pendulum, the period of which is equal to the period of the given physical pendulum. The reduced length is calculated as follows:

where I is the moment of inertia about the suspension point, m is the mass, a is the distance from the suspension point to the center of mass.

An oscillatory circuit is an oscillator, which is an electrical circuit containing a connected inductor and a capacitor. Current (and voltage) oscillations can be excited in such a circuit. An oscillatory circuit is the simplest system in which free electromagnetic oscillations can occur.

the resonant frequency of the circuit is determined by the so-called Thomson formula:

Parallel oscillatory circuit

Let a capacitor of capacity C be charged to a voltage. The energy stored in the capacitor is

The magnetic energy concentrated in the coil is maximum and equal to

Where L is the inductance of the coil, is the maximum value of the current.

Energy of harmonic vibrations

During mechanical vibrations, an oscillating body (or material point) has kinetic and potential energy. Kinetic energy of the body W:

Total energy in the circuit:

Electromagnetic waves carry energy. When waves propagate, a flow of electromagnetic energy arises. If we single out the area S, oriented perpendicular to the direction of wave propagation, then in a short time Δt, the energy ΔWem will flow through the area, equal to ΔWem = (we + wm)υSΔt

13. Addition of harmonic oscillations of the same direction and the same frequency

An oscillating body can take part in several oscillatory processes, then the resulting oscillation should be found, in other words, the oscillations must be added. In this section, we will add harmonic oscillations of the same direction and the same frequency

using the rotating amplitude vector method, we construct graphically the vector diagrams of these oscillations (Fig. 1). Tax as the vectors A1 and A2 rotate with the same angular velocity ω0, then the phase difference (φ2 - φ1) between them will remain constant. Hence, the equation of the resulting oscillation will be (1)

In formula (1), the amplitude A and the initial phase φ are respectively determined by the expressions

This means that the body, participating in two harmonic oscillations of the same direction and the same frequency, also performs a harmonic oscillation in the same direction and with the same frequency as the summed oscillations. The amplitude of the resulting oscillation depends on the phase difference (φ2 - φ1) of the added oscillations.

Addition of harmonic oscillations of the same direction with close frequencies

Let the amplitudes of the added oscillations be equal to A, and the frequencies be equal to ω and ω + Δω, and Δω<<ω. Выберем начало отсчета так, чтобы начальные фазы обоих колебаний были равны нулю:

Adding these expressions and taking into account that in the second factor Δω/2<<ω, получим

Periodic changes in the amplitude of oscillations that occur when two harmonic oscillations of the same direction with close frequencies are added are called beats.

Beats arise from the fact that one of the two signals constantly lags behind the other in phase, and at those moments when the oscillations occur in phase, the total signal is amplified, and at those moments when the two signals are out of phase, they cancel each other out. These moments periodically replace each other as the backlog increases.

Beat oscillation chart

Let us find the result of adding two harmonic oscillations of the same frequency ω, which occur in mutually perpendicular directions along the x and y axes. For simplicity, we choose the origin of reference so that the initial phase of the first oscillation is equal to zero, and write it in the form (1)

where α is the phase difference of both oscillations, A and B are equal to the amplitudes of the added oscillations. The trajectory equation of the resulting oscillation will be determined by excluding the time t from formulas (1). Writing the summed oscillations as

and replacing in the second equation by and by , we find, after simple transformations, the equation of an ellipse whose axes are arbitrarily oriented relative to the coordinate axes: (2)

Since the trajectory of the resulting oscillation has the shape of an ellipse, such oscillations are called elliptically polarized.

The dimensions of the axes of the ellipse and its orientation depend on the amplitudes of the added oscillations and the phase difference α. Let us consider some special cases that are of physical interest to us:

1) α = mπ (m=0, ±1, ±2, ...). In this case, the ellipse becomes a straight line segment (3)

where the plus sign corresponds to zero and even values ​​of m (Fig. 1a), and the minus sign corresponds to odd values ​​of m (Fig. 2b). The resulting oscillation is a harmonic oscillation with frequency ω and amplitude, which occurs along the straight line (3), making an angle with the x-axis. In this case, we are dealing with linearly polarized oscillations;

2) α = (2m+1)(π/2) (m=0, ± 1, ±2,...). In this case, the equation will look like

Lissajous figures are closed trajectories drawn by a point that simultaneously performs two harmonic oscillations in two mutually perpendicular directions. First studied by the French scientist Jules Antoine Lissajous. The shape of the figures depends on the relationship between the periods (frequencies), phases and amplitudes of both oscillations. In the simplest case of equality of both periods, the figures are ellipses, which, with a phase difference of 0 or degenerate into line segments, and with a phase difference of P / 2 and equality of amplitudes, turn into a circle. If the periods of both oscillations do not exactly coincide, then the phase difference changes all the time, as a result of which the ellipse is deformed all the time. Lissajous figures are not observed for significantly different periods. However, if the periods are related as integers, then after a time interval equal to the smallest multiple of both periods, the moving point returns to the same position again - Lissajous figures of a more complex form are obtained. The Lissajous figures are inscribed in a rectangle whose center coincides with the origin of coordinates, and the sides are parallel to the coordinate axes and located on both sides of them at distances equal to the oscillation amplitudes.

where A, B - oscillation amplitudes, a, b - frequencies, δ - phase shift

14. Damped oscillations occur in a closed mechanical system

In which there are energy losses to overcome forces

resistance (β ≠ 0) or in a closed oscillatory circuit, in

where the presence of resistance R leads to the loss of vibration energy on

heating of conductors (β ≠ 0).

In this case, the general differential oscillation equation (5.1)

takes the form: x′′ + 2βx′ + ω0 x = 0 .

The logarithmic damping decrement χ is a physical quantity reciprocal to the number of oscillations after which the amplitude A decreases by a factor of e.

APERIODIC PROCESS-transient process in dynamic. system, for which the output value, characterizing the transition of the system from one state to another, either monotonically tends to a steady value, or has one extremum (see Fig.). Theoretically, it can last an infinitely long time. A. p. take place, for example, in automatic systems. management.

Graphs of aperiodic processes of changing the parameter x(t) of the system in time: xust - steady state (limiting) value of the parameter

The smallest active resistance of the circuit, at which the process is aperiodic, is called critical resistance

It is also such a resistance at which the mode of free undamped oscillations is realized in the circuit.

15. Oscillations that occur under the action of an external periodically changing force or an external periodically changing emf are called forced mechanical and forced electromagnetic oscillations, respectively.

The differential equation will take the following form:

q′′ + 2βq′ + ω0 q = cos(ωt) .

Resonance (fr. resonance, from lat. resono - I respond) is a phenomenon of a sharp increase in the amplitude of forced oscillations, which occurs when the frequency of an external influence approaches certain values ​​(resonant frequencies) determined by the properties of the system. An increase in amplitude is only a consequence of resonance, and the reason is the coincidence of the external (exciting) frequency with the internal (natural) frequency of the oscillatory system. With the help of the resonance phenomenon, even very weak periodic oscillations can be isolated and/or enhanced. Resonance is a phenomenon that, at a certain frequency of the driving force, the oscillatory system is especially responsive to the action of this force. The degree of responsiveness in oscillation theory is described by a quantity called the quality factor. The phenomenon of resonance was first described by Galileo Galilei in 1602 in works devoted to the study of pendulums and musical strings.

The mechanical resonant system best known to most people is the usual swing. If you push the swing according to its resonant frequency, the range of motion will increase, otherwise the motion will die out. The resonant frequency of such a pendulum with sufficient accuracy in the range of small displacements from the equilibrium state can be found by the formula:

where g is the free fall acceleration (9.8 m/s² for the Earth's surface), and L is the length from the pendulum's suspension point to its center of mass. (A more precise formula is rather complicated, and involves an elliptic integral). It is important that the resonant frequency does not depend on the mass of the pendulum. It is also important that you cannot swing the pendulum at multiple frequencies (higher harmonics), but this can be done at frequencies equal to fractions of the fundamental (lower harmonics).

Amplitude and phase of forced oscillations.

Consider the dependence of the amplitude A of forced oscillations on the frequency ω (8.1)

From formula (8.1) it follows that the displacement amplitude A has a maximum. To determine the resonant frequency ωres - the frequency at which the displacement amplitude A reaches its maximum - you need to find the maximum of the function (1), or, what is the same, the minimum of the radical expression. Differentiating the radical expression with respect to ω and equating it to zero, we obtain the condition that determines ωres:

This equality holds for ω=0, ± , for which only a positive value has a physical meaning. Therefore, the resonant frequency (8.2)

The phenomenon of electromagnetic induction is used primarily to convert mechanical energy into electric current energy. For this purpose, apply alternators(induction generators). The simplest alternating current generator is a wire frame rotating uniformly with an angular velocity w= const in a uniform magnetic field with induction AT(Fig. 4.5). The flux of magnetic induction penetrating a frame with an area S, is equal to

With uniform rotation of the frame, the angle of rotation , where is the rotation frequency. Then

According to the law of electromagnetic induction, the EMF induced in the frame at
her rotation,

If a load (electricity consumer) is connected to the frame clamps using a brush-contact apparatus, then alternating current will flow through it.

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.

Consider the principle of operation of a two-winding transformer (Fig. 4.7). When an alternating current passes through the primary winding, an alternating magnetic field arises around it with induction AT, whose flow is also variable

The core of the transformer serves to direct the magnetic flux (the magnetic resistance of the air is high). A variable magnetic flux, closing along the core, induces a variable EMF in each of the windings:

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 vacuum () the speed of propagation of electromagnetic waves coincides with the speed of light; in matter, so the speed of propagation of electromagnetic waves in matter is always less than in vacuum.

The phenomenon of electromagnetic induction is a phenomenon that consists in the occurrence of an electromotive force or voltage in a body that is in a magnetic field that is constantly changing. The electromotive force as a result of electromagnetic induction also arises if the body moves in a static and inhomogeneous magnetic field or rotates in a magnetic field so that its lines intersecting a closed contour change.

Induced electric current

The concept of "induction" means the occurrence of a process as a result of the influence of another process. For example, an electric current can be induced, that is, it can appear as a result of exposing a conductor to a magnetic field in a special way. Such an electric current is called induced. The conditions for the formation of an electric current as a result of the phenomenon of electromagnetic induction are discussed later in the article.

The concept of a magnetic field

Before starting to study the phenomenon of electromagnetic induction, it is necessary to understand what a magnetic field is. In simple terms, a magnetic field is a region of space in which a magnetic material exhibits its magnetic effects and properties. This region of space can be depicted using lines called magnetic field lines. The number of these lines represents a physical quantity called magnetic flux. The magnetic field lines are closed, they start at the north pole of the magnet and end at the south.

The magnetic field has the ability to act on any materials that have magnetic properties, for example, iron conductors of electric current. This field is characterized by magnetic induction, which is denoted B and is measured in tesla (T). A magnetic induction of 1 T is a very strong magnetic field that acts with a force of 1 newton on a point charge of 1 coulomb, which flies perpendicular to the magnetic field lines at a speed of 1 m / s, that is, 1 T = 1 N * s / ( m*Cl).

Who discovered the phenomenon of electromagnetic induction?

Electromagnetic induction, on the principle of operation of which many modern devices are based, was discovered in the early 30s of the 19th century. The discovery of induction is usually attributed to Michael Faraday (discovery date - August 29, 1831). The scientist was based on the results of the experiments of the Danish physicist and chemist Hans Oersted, who discovered that a conductor through which an electric current flows creates a magnetic field around itself, that is, it begins to exhibit magnetic properties.

Faraday, in turn, discovered the opposite of the phenomenon discovered by Oersted. He noticed that a changing magnetic field, which can be created by changing the parameters of the electric current in the conductor, leads to the appearance of a potential difference at the ends of any current conductor. If these ends are connected, for example, through an electric lamp, then an electric current will flow through such a circuit.

As a result, Faraday discovered a physical process, as a result of which an electric current appears in a conductor due to a change in the magnetic field, which is the phenomenon of electromagnetic induction. At the same time, for the formation of an induced current, it does not matter what moves: the magnetic field or itself can be easily shown if an appropriate experiment is carried out on the phenomenon of electromagnetic induction. So, having placed the magnet inside the metal spiral, we begin to move it. If you connect the ends of the spiral through any indicator of electric current into a circuit, you can see the appearance of current. Now you should leave the magnet alone and move the spiral up and down relative to the magnet. The indicator will also show the existence of current in the circuit.

Faraday experiment

Faraday's experiments consisted of working with a conductor and a permanent magnet. Michael Faraday first discovered that when a conductor moves inside a magnetic field, a potential difference arises at its ends. The moving conductor begins to cross the lines of the magnetic field, which simulates the effect of changing this field.

The scientist found that the positive and negative signs of the resulting potential difference depend on the direction in which the conductor moves. For example, if the conductor is raised in a magnetic field, then the resulting potential difference will have a +- polarity, but if this conductor is lowered, then we will already get a -+ polarity. These changes in the sign of the potentials, the difference of which is called the electromotive force (EMF), lead to the appearance of an alternating current in a closed circuit, that is, a current that constantly changes its direction to the opposite.

Features of electromagnetic induction discovered by Faraday

Knowing who discovered the phenomenon of electromagnetic induction and why an induced current occurs, we will explain some of the features of this phenomenon. So, the faster you move the conductor in a magnetic field, the greater the value of the induced current in the circuit will be. Another feature of the phenomenon is as follows: the greater the magnetic induction of the field, that is, the stronger this field, the greater the potential difference it can create when moving the conductor in the field. If the conductor is at rest in a magnetic field, no EMF arises in it, since there is no change in the lines of magnetic induction crossing the conductor.

Electric current direction and left hand rule

To determine the direction in the conductor of the electric current created as a result of the phenomenon of electromagnetic induction, you can use the so-called left-hand rule. It can be formulated as follows: if the left hand is placed so that the lines of magnetic induction, which begin at the north pole of the magnet, enter the palm, and the protruding thumb is directed in the direction of movement of the conductor in the field of the magnet, then the remaining four fingers of the left hand will indicate the direction of movement induced current in the conductor.

There is another version of this rule, it is as follows: if the index finger of the left hand is directed along the lines of magnetic induction, and the protruding thumb is directed in the direction of the conductor, then the middle finger turned 90 degrees to the palm will indicate the direction of the appeared current in the conductor.

The phenomenon of self-induction

Hans Christian Oersted discovered the existence of a magnetic field around a current-carrying conductor or coil. The scientist also found that the characteristics of this field are directly related to the strength of the current and its direction. If the current in the coil or conductor is variable, then it will generate a magnetic field that will not be stationary, that is, it will change. In turn, this alternating field will lead to the appearance of an induced current (the phenomenon of electromagnetic induction). The movement of the induction current will always be opposite to the alternating current circulating through the conductor, that is, it will resist each change in the direction of the current in the conductor or coil. This process is called self-induction. The difference in electrical potential created in this case is called the self-inductance emf.

Note that the phenomenon of self-induction occurs not only when the direction of the current changes, but also with any change in it, for example, with an increase due to a decrease in the resistance in the circuit.

To physically describe the resistance exerted by any change in current in a circuit due to self-induction, the concept of inductance was introduced, which is measured in Henry (in honor of the American physicist Joseph Henry). One henry is such an inductance for which, when the current changes by 1 ampere in 1 second, an EMF arises in the process of self-induction, equal to 1 volt.

Alternating current

When the inductor begins to rotate in a magnetic field, as a result of the phenomenon of electromagnetic induction, it creates an induced current. This electric current is variable, that is, it systematically changes its direction.

Alternating current is more common than direct current. So, many devices that operate from the central electrical network use this type of current. Alternating current is easier to induce and transport than direct current. As a rule, the frequency of household alternating current is 50-60 Hz, that is, in 1 second its direction changes 50-60 times.

The geometric representation of alternating current is a sinusoidal curve that describes the dependence of voltage on time. The full period of the sinusoidal curve for household current is approximately 20 milliseconds. According to the thermal effect, alternating current is similar to direct current, the voltage of which is U max /√2, where U max is the maximum voltage on the sinusoidal alternating current curve.

The use of electromagnetic induction in technology

The discovery of the phenomenon of electromagnetic induction produced a real boom in the development of technology. Before this discovery, humans were only able to produce electricity in limited quantities using electric batteries.

Currently, this physical phenomenon is used in electrical transformers, in heaters that convert induced current into heat, as well as in electric motors and car generators.