Definition of electromagnetic field. What are electromagnetic fields (EMF)

An electromagnetic field is a type of matter that arises around moving charges. For example, around a conductor carrying current. The electromagnetic field consists of two components: electric and magnetic field. They cannot exist independently of each other. One thing begets another. When the electric field changes, a magnetic field immediately appears.

Electromagnetic wave propagation speed V=C/EM

Where e And m respectively, the magnetic and dielectric constants of the medium in which the wave propagates.
An electromagnetic wave in a vacuum travels at the speed of light, that is, 300,000 km/s. Since the dielectric and magnetic permeability of a vacuum are considered equal to 1.

When the electric field changes, a magnetic field appears. Since the electric field that caused it is not constant (that is, it changes over time), the magnetic field will also be variable.

A changing magnetic field in turn generates an electric field, and so on. Thus, for the subsequent field (it does not matter whether it is electric or magnetic), the source will be the previous field, and not the original source, that is, a conductor with current.

Thus, even after turning off the current in the conductor, the electromagnetic field will continue to exist and spread in space.

An electromagnetic wave propagates in space in all directions from its source. You can imagine turning on a light bulb, the rays of light from it spread in all directions.

An electromagnetic wave, when propagating, transfers energy in space. The stronger the current in the conductor that causes the field, the greater the energy transferred by the wave. Also, the energy depends on the frequency of the emitted waves; if it increases by 2,3,4 times, the wave energy will increase by 4,9,16 times, respectively. That is, the energy of wave propagation is proportional to the square of the frequency.

The best conditions for wave propagation are created when the length of the conductor is equal to the wavelength.

The magnetic and electric lines of force will fly mutually perpendicular. Magnetic lines of force surround a current-carrying conductor and are always closed.
Electrical lines of force go from one charge to another.

An electromagnetic wave is always a transverse wave. That is, the lines of force, both magnetic and electric, lie in a plane perpendicular to the direction of propagation.

Electromagnetic field strength is a strength characteristic of the field. Also, tension is a vector quantity, that is, it has a beginning and a direction.
The field strength is directed tangentially to the lines of force.

Since the electric and magnetic field strengths are perpendicular to each other, there is a rule by which the direction of wave propagation can be determined. When the screw rotates along the shortest path from the electric field strength vector to the magnetic field strength vector, the forward movement of the screw will indicate the direction of wave propagation.

Sources of electromagnetic fields (EMF) are extremely diverse - these are power transmission and distribution systems (power lines, transformer and distribution substations) and devices that consume electricity (electric motors, electric stoves, electric heaters, refrigerators, televisions, video display terminals, etc.).

Sources that generate and transmit electromagnetic energy include radio and television broadcast stations, radar installations and radio communication systems, a wide variety of technological installations in industry, medical devices and equipment (devices for diathermy and inductothermy, UHF therapy, devices for microwave therapy and etc.).

The working contingent and population may be exposed to isolated electric or magnetic field components or a combination of both. Depending on the relationship of the exposed person to the source of radiation, it is customary to distinguish between several types of exposure - professional, non-professional, exposure at home and exposure for therapeutic purposes. Occupational exposure is characterized by a variety of generation modes and options for exposure to electromagnetic fields (irradiation in the near zone, in the induction zone, general and local, combined with the action of other unfavorable factors in the working environment). In conditions of non-occupational exposure, the most typical is general exposure, in most cases in the wave zone.

Electromagnetic fields generated by certain sources can affect the entire body of a working person (general exposure) or a separate part of the body (local exposure). In this case, exposure can be isolated (from one EMF source), combined (from two or more EMF sources of the same frequency range), mixed (from two or more EMF sources of different frequency ranges), as well as combined (under conditions of simultaneous exposure to EMF and other unfavorable physical factors of the working environment) exposure.

An electromagnetic wave is an oscillatory process associated with interconnected electric and magnetic fields varying in space and time.

An electromagnetic field is the area of ​​propagation of electromagnetic

Characteristics of electromagnetic waves. An electromagnetic field is characterized by a radiation frequency f, measured in hertz, or a wavelength X, measured in meters. An electromagnetic wave propagates in a vacuum at the speed of light (3,108 m/s), and the relationship between the length and frequency of the electromagnetic wave is determined by the relationship

where c is the speed of light.

The speed of propagation of waves in air is close to the speed of their propagation in vacuum.

An electromagnetic field has energy, and an electromagnetic wave, propagating in space, transfers this energy. The electromagnetic field has electric and magnetic components (Table No. 35).

Electric field strength E is a characteristic of the electrical component of the EMF, the unit of measurement of which is V/m.

Magnetic field strength H (A/m) is a characteristic of the magnetic component of the EMF.

Energy flux density (EFD) is the energy of an electromagnetic wave transferred by an electromagnetic wave per unit time through a unit area. The unit of measurement for PES is W/m.

For certain ranges of electromagnetic radiation - EMR (light range, laser radiation) other characteristics have been introduced.

Classification of electromagnetic fields. The frequency range and length of the electromagnetic wave make it possible to classify the electromagnetic field into visible light (light waves), infrared (thermal) and ultraviolet radiation, the physical basis of which is electromagnetic waves. These types of short-wave radiation have a specific effect on humans.

The physical basis of ionizing radiation is also made up of electromagnetic waves of very high frequencies, which have high energy sufficient to ionize the molecules of the substance in which the wave propagates (Table No. 36).

The radio frequency range of the electromagnetic spectrum is divided into four frequency ranges: low frequencies (LF) - less than 30 kHz, high frequencies (HF) - 30 kHz...30 MHz, ultra high frequencies (UHF) - 30...300 MHz, ultra high frequencies ( Microwave) - 300 MHz.750 GHz.

A special type of electromagnetic radiation (EMR) is laser radiation (LR), generated in the wavelength range 0.1...1000 microns. The peculiarity of LR is its monochromaticity (strictly one wavelength), coherence (all radiation sources emit waves in the same phase), and sharp beam directionality (small beam divergence).

Conventionally, non-ionizing radiation (fields) can include electrostatic fields (ESF) and magnetic fields (MF).

An electrostatic field is a field of stationary electric charges that interacts between them.

Static electricity is a set of phenomena associated with the emergence, conservation and relaxation of a free electric charge on the surface or in the volume of dielectrics or on insulated conductors.

The magnetic field can be constant, pulsed, alternating.

Depending on the sources of formation, electrostatic fields can exist in the form of an electrostatic field itself, formed in various types of power plants and during electrical processes. In industry, ESPs are widely used for electrogas purification, electrostatic separation of ores and materials, and electrostatic application of paints and polymers. Manufacturing, testing,

transportation and storage of semiconductor devices and integrated circuits, grinding and polishing of cases for radio and television receivers,

technological processes associated with the use of dielectric

materials, as well as the premises of computer centers where multiplying computer technology is concentrated are characterized by the formation

electrostatic fields. Electrostatic charges and the electrostatic fields they create can arise when dielectric liquids and some bulk materials move through pipelines, when dielectric liquids are poured, or when film or paper is rolled.

Electromagnets, solenoids, capacitor-type installations, cast and cermet magnets are accompanied by the appearance of magnetic fields.

In electromagnetic fields, three zones are distinguished, which are formed at different distances from the source of electromagnetic radiation.

Induction zone (near zone) - covers the interval from the radiation source to a distance equal to approximately V2n ~ V6. In this zone, the electromagnetic wave has not yet been formed and therefore the electric and magnetic fields are not interconnected and act independently (first zone).

The interference zone (intermediate zone) is located at distances from approximately V2n to 2lX. In this zone, electromagnetic waves are formed and a person is affected by electric and magnetic fields, as well as an energy impact (second zone).

Wave zone (far zone) - located at distances greater than 2lX. In this zone, an electromagnetic wave is formed, and the electric and magnetic fields are interconnected. A person in this zone is affected by wave energy (third zone).

The effect of the electromagnetic field on the body. The biological and pathophysiological effect of electromagnetic fields on the body depends on the frequency range, the intensity of the influencing factor, the duration of irradiation, the nature of the radiation and the irradiation mode. The effect of EMF on the body depends on the pattern of propagation of radio waves in material environments, where the absorption of electromagnetic wave energy is determined by the frequency of electromagnetic oscillations, electrical and magnetic properties of the medium.

As is known, the leading indicator characterizing the electrical properties of body tissues is their dielectric and magnetic permeability. In turn, differences in the electrical properties of tissues (dielectric and magnetic permeability, resistivity) are associated with the content of free and bound water in them. All biological tissues, according to dielectric constant, are divided into two groups: tissues with a high water content - over 80% (blood, muscles, skin, brain tissue, liver and spleen tissue) and tissues with a relatively low water content (fat, bone). The absorption coefficient in tissues with high water content, at the same field strength, is 60 times higher than in tissues with low water content. Therefore, the depth of penetration of electromagnetic waves into tissues with a low water content is 10 times greater than in tissues with a high water content.

Thermal and athermic effects underlie the mechanisms of the biological action of electromagnetic waves. The thermal effect of EMF is characterized by selective heating of individual organs and tissues and an increase in overall body temperature. Intense EMF irradiation can cause destructive changes in tissues and organs, however, acute forms of damage are extremely rare and their occurrence is most often associated with emergency situations when safety precautions are violated.

Chronic forms of radio wave injuries, their symptoms and course do not have strictly specific manifestations. However, they are characterized by the development of asthenic conditions and vegetative disorders, mainly with

aspects of the cardiovascular system. Along with general asthenia, accompanied by weakness, increased fatigue, restless sleep, patients experience headache, dizziness, psycho-emotional lability, pain in the heart, increased sweating, and decreased appetite. Signs of acrocyanosis, regional hyperhidrosis, cold hands and feet, tremor of the fingers, lability of pulse and blood pressure with a tendency to bradycardia and hypotension develop; Dysfunction in the pituitary-adrenal cortex system leads to changes in the secretion of thyroid and sex hormones.

One of the few specific lesions caused by exposure to electromagnetic radiation in the radio frequency range is the development of cataracts. In addition to cataracts, when exposed to high-frequency electromagnetic waves, keratitis and damage to the corneal stroma can develop.

Infrared (thermal) radiation, light radiation at high energies, as well as high-level ultraviolet radiation, with acute exposure, can lead to dilation of capillaries, burns of the skin and organs of vision. Chronic irradiation is accompanied by changes in skin pigmentation, the development of chronic conjunctivitis and clouding of the eye lens. Ultraviolet radiation at low levels is useful and necessary for humans, as it enhances metabolic processes in the body and the synthesis of the biologically active form of vitamin D.

The effect of laser radiation on a person depends on the intensity of the radiation, wavelength, nature of the radiation and exposure time. In this case, local and general damage to certain tissues of the human body is distinguished. The target organ in this case is the eye, which is easily damaged, the transparency of the cornea and lens is impaired, and damage to the retina is possible. Laser scanning, especially in the infrared range, can penetrate tissue to a considerable depth, affecting internal organs. Long-term exposure to laser radiation of even low intensity can lead to various functional disorders of the nervous, cardiovascular systems, endocrine glands, blood pressure, increased fatigue, and decreased performance.

Hygienic regulation of electromagnetic fields. According to regulatory documents: SanPiN “Sanitary and epidemiological requirements for the operation of radio-electronic equipment with working conditions with sources of electromagnetic radiation” No. 225 dated April 10, 2007, Ministry of Health of the Republic of Kazakhstan; SanPiN “Sanitary rules and standards for the protection of the population from the effects of electromagnetic fields created by radio engineering objects” No. 3.01.002-96 of the Ministry of Health of the Republic of Kazakhstan; MU

“Guidelines for the implementation of state sanitary supervision of objects with sources of electromagnetic fields (EMF) of the non-ionizing part of the spectrum” No. 1.02.018/u-94 of the Ministry of Health of the Republic of Kazakhstan; MU "Methodological recommendations for laboratory monitoring of sources of electromagnetic fields of the non-ionizing part of the spectrum (EMF) during state sanitary supervision" No. 1.02.019/r-94 The Ministry of Health of the Republic of Kazakhstan regulates the intensity of electromagnetic fields of radio frequencies at personnel workplaces,
carrying out work with EMF sources and requirements for monitoring, and irradiation with an electric field is also regulated, both in terms of intensity and duration of action.

The frequency range of radio frequencies of electromagnetic fields (60 kHz - 300 MHz) is estimated by the strength of the electric and magnetic components of the field; in the frequency range 300 MHz - 300 GHz - by the surface radiation energy flux density and the energy load (EL) created by it. The total energy flow passing through a unit of irradiated surface during the action time (T), and expressed by the product of PES T, represents the energy load.

In cases where the time of exposure to EMFs on personnel does not exceed 50% of the working time, levels higher than those specified are allowed, but not more than 2 times.

Standardization and hygienic assessment of permanent magnetic fields (PMF) in industrial premises and workplaces (Table No. 37) is carried out differentiated, depending on the time of exposure to the employee during the work shift and taking into account the conditions of general or local exposure.

The PMP hygienic standards (Table No. 38), developed by the International Committee on Non-Ionizing Radiation, which operates under the International Radiation Protection Association, are also widely used.

Instructions

Take two batteries and connect them with electrical tape. Connect the batteries so that their ends are different, that is, the plus is opposite the minus and vice versa. Use paper clips to attach a wire to the end of each battery. Next, place one of the paper clips on top of the batteries. If the paperclip does not reach the center of each paperclip, it may need to be bent to the correct length. Secure the structure with tape. Make sure the ends of the wires are clear and the edge of the paperclip reaches the center of each battery. Connect the batteries from the top, do the same on the other side.

Take copper wire. Leave about 15 centimeters of the wire straight, and then start wrapping it around the glass cup. Make about 10 turns. Leave another 15 centimeters straight. Connect one of the wires from the power supply to one of the free ends of the resulting copper coil. Make sure the wires are well connected to each other. When connected, the circuit produces a magnetic field. Connect the other wire of the power supply to the copper wire.

When current flows through the coil, the coil placed inside will be magnetized. Paper clips will stick together, and parts of a spoon or fork or screwdriver will become magnetized and attract other metal objects while current is applied to the coil.

note

The coil may be hot. Make sure there are no flammable substances nearby and be careful not to burn your skin.

Helpful advice

The most easily magnetized metal is iron. When checking the field, do not select aluminum or copper.

In order to make an electromagnetic field, you need to make its source radiate. At the same time, it must produce a combination of two fields, electric and magnetic, which can propagate in space, generating each other. An electromagnetic field can propagate in space in the form of an electromagnetic wave.

You will need

• - insulated wire;
• - nail;
• - two conductors;
• - Ruhmkorff coil.

Instructions

Take an insulated wire with low resistance, copper is best. Wind it around a steel core; a regular nail 100 mm long (one hundred square meters) will do. Connect the wire to a power source; a regular battery will do. Electricity will arise field, which will generate an electric current in it.

Directed movement of charged (electric current) will in turn give rise to magnetic field, which will be concentrated in a steel core, with a wire wound around it. The core transforms and attracts ferromagnets (nickel, cobalt, etc.). The resulting field can be called electromagnetic, since electric field magnetic.

To obtain a classical electromagnetic field, it is necessary that both electric and magnetic field changed over time, then electrical field will generate magnetic and vice versa. To do this, moving charges need to be accelerated. The easiest way to do this is to make them hesitate. Therefore, to obtain an electromagnetic field, it is enough to take a conductor and plug it into a regular household network. But it will be so small that it will not be possible to measure it with instruments.

To obtain a sufficiently powerful magnetic field, make a Hertz vibrator. To do this, take two straight identical conductors and fasten them so that the gap between them is 7 mm. This will be an open oscillatory circuit, with low electrical capacity. Connect each of the conductors to Ruhmkorff clamps (it allows you to receive high voltage pulses). Connect the circuit to the battery. Discharges will begin in the spark gap between the conductors, and the vibrator itself will become a source of an electromagnetic field.

Video on the topic

The introduction of new technologies and the widespread use of electricity has led to the emergence of artificial electromagnetic fields, which most often have a harmful effect on humans and the environment. These physical fields arise where there are moving charges.

The nature of the electromagnetic field

The electromagnetic field is a special type of matter. It occurs around conductors along which electric charges move. The force field consists of two independent fields - magnetic and electric, which cannot exist in isolation from one another. When an electric field arises and changes, it invariably generates a magnetic field.

One of the first to study the nature of alternating fields in the middle of the 19th century was James Maxwell, who is credited with creating the theory of the electromagnetic field. The scientist showed that electric charges moving with acceleration create an electric field. Changing it generates a field of magnetic forces.

The source of an alternating magnetic field can be a magnet if it is set in motion, as well as an electric charge that oscillates or moves with acceleration. If a charge moves at a constant speed, then a constant current flows through the conductor, which is characterized by a constant magnetic field. Propagating in space, the electromagnetic field transfers energy, which depends on the magnitude of the current in the conductor and the frequency of the emitted waves.

Impact of electromagnetic field on humans

The level of all electromagnetic radiation created by man-made technical systems is many times higher than the natural radiation of the planet. This is a thermal effect that can lead to overheating of body tissues and irreversible consequences. For example, prolonged use of a mobile phone, which is a source of radiation, can lead to an increase in the temperature of the brain and the lens of the eye.

Electromagnetic fields generated when using household appliances can cause the appearance of malignant tumors. This especially applies to children's bodies. A person's prolonged presence near a source of electromagnetic waves reduces the efficiency of the immune system and leads to heart and vascular diseases.

Of course, it is impossible to completely abandon the use of technical means that are a source of electromagnetic fields. But you can use the simplest preventive measures, for example, use your phone only with a headset, and do not leave appliance cords in electrical outlets after using equipment. In everyday life, it is recommended to use extension cords and cables that have protective shielding.

Shmelev V.E., Sbitnev S.A.

"THEORETICAL FUNDAMENTALS OF ELECTRICAL ENGINEERING"

"ELECTROMAGNETIC FIELD THEORY"

Chapter 1. Basic concepts of electromagnetic field theory

§ 1.1. Definition of the electromagnetic field and its physical quantities.
Mathematical apparatus of the theory of electromagnetic field

Electromagnetic field(EMF) is a type of matter that exerts a force on charged particles and is determined at all points by two pairs of vector quantities that characterize its two sides - electric and magnetic fields.

Electric field- this is a component of EMF, which is characterized by the effect on an electrically charged particle with a force proportional to the charge of the particle and independent of its speed.

A magnetic field is a component of EMF, which is characterized by the effect on a moving particle with a force proportional to the charge of the particle and its speed.

The basic properties and methods of calculating EMFs studied in the course of theoretical foundations of electrical engineering involve a qualitative and quantitative study of EMFs found in electrical, electronic and biomedical devices. For this purpose, the equations of electrodynamics in integral and differential forms are most suitable.

The mathematical apparatus of electromagnetic field theory (TEMF) is based on scalar field theory, vector and tensor analysis, as well as differential and integral calculus.

Control questions

1. What is an electromagnetic field?

2. What are called electric and magnetic fields?

3. What is the mathematical apparatus of the electromagnetic field theory based on?

§ 1.2. Physical quantities characterizing EMF

Electric field strength vector at the point Q is the vector of force acting on an electrically charged stationary particle placed at a point Q, if this particle has a unit positive charge.

According to this definition, the electric force acting on a point charge q is equal to:

Where E measured in V/m.

The magnetic field is characterized vector of magnetic induction. Magnetic induction at some observation point Q is a vector quantity whose modulus is equal to the magnetic force acting on a charged particle located at a point Q, having a unit charge and moving with a unit speed, and the vectors of force, speed, magnetic induction, as well as the charge of the particle satisfy the condition

.

The magnetic force acting on a curved conductor carrying current can be determined by the formula

.

A straight conductor, if it is in a uniform field, is acted upon by the following magnetic force

.

In all the latest formulas B - magnetic induction, which is measured in teslas (T).

1 T is a magnetic induction in which a magnetic force equal to 1 N acts on a straight conductor with a current of 1A, if the lines of magnetic induction are directed perpendicular to the conductor with the current, and if the length of the conductor is 1 m.

In addition to the electric field strength and magnetic induction, the following vector quantities are considered in the theory of the electromagnetic field:

1) electrical induction D (electrical displacement), which is measured in C/m 2,

EMF vectors are functions of space and time:

Where Q- observation point, t- moment of time.

If the observation point Q is in a vacuum, then the following relations hold between the corresponding pairs of vector quantities

where is the absolute dielectric constant of vacuum (basic electrical constant), =8.85419*10 -12;

Absolute magnetic permeability of vacuum (basic magnetic constant); = 4π*10 -7 .

Control questions

1. What is electric field strength?

2. What is magnetic induction called?

3. What is the magnetic force acting on a moving charged particle?

4. What is the magnetic force acting on a current-carrying conductor?

5. What vector quantities are characterized by the electric field?

6. What vector quantities are characterized by a magnetic field?

§ 1.3. Electromagnetic field sources

Sources of EMF are electric charges, electric dipoles, moving electric charges, electric currents, magnetic dipoles.

The concepts of electric charge and electric current are given in the physics course. Electric currents are of three types:

1. Conduction currents.

2. Displacement currents.

3. Transfer currents.

Conduction current- the speed of passage of moving charges of an electrically conductive body through a certain surface.

Bias current- the rate of change of the electric displacement vector flow through a certain surface.

.

Transfer current characterized by the following expression

Where v - speed of transfer of bodies through the surface S; n - vector of the unit normal to the surface; - linear charge density of bodies flying through the surface in the direction of the normal; ρ - volume density of electric charge; ρ v - transfer current density.

Electric dipole called a pair of point charges + q And - q, located at a distance l from each other (Fig. 1).

A point electric dipole is characterized by the vector of the electric dipole moment:

Magnetic dipole called a flat circuit with electric current I. A magnetic dipole is characterized by the vector of the magnetic dipole moment

Where S - vector of the area of ​​a flat surface stretched over a current-carrying circuit. Vector S directed perpendicular to this flat surface, and, when viewed from the end of the vector S , then movement along the contour in the direction coinciding with the direction of the current will occur counterclockwise. This means that the direction of the dipole magnetic moment vector is related to the direction of the current according to the right-hand screw rule.

Atoms and molecules of matter are electric and magnetic dipoles, therefore each point of a material type in the EMF can be characterized by the volumetric density of the electric and magnetic dipole moment:

P - electrical polarization of the substance:

M - magnetization of the substance:

Electrical polarization of matter is a vector quantity equal to the volumetric density of the electric dipole moment at some point of a real body.

Magnetization of a substance is a vector quantity equal to the volumetric density of the magnetic dipole moment at some point of a material body.

Electrical bias is a vector quantity, which for any observation point, regardless of whether it is in a vacuum or in matter, is determined from the relation:

(for vacuum or substance),

(for vacuum only).

Magnetic field strength- a vector quantity, which for any observation point, regardless of whether it is in a vacuum or in a substance, is determined from the relation:

,

where the magnetic field strength is measured in A/m.

In addition to polarization and magnetization, there are other volumetrically distributed sources of EMF:

- volumetric charge density ; ,

where the volumetric charge density is measured in C/m3;

- electric current density vector, whose normal component is equal to

More generally, the current flowing through an open surface S, is equal to the current density vector flux through this surface:

where the electric current density vector is measured in A/m 2.

Control questions

1. What are the sources of the electromagnetic field?

2. What is conduction current?

3. What is bias current?

4. What is transfer current?

5. What is an electric dipole and an electric dipole moment?

6. What is a magnetic dipole and magnetic dipole moment?

7. What is called the electrical polarization and magnetization of a substance?

8. What is called electrical displacement?

9. What is magnetic field strength called?

10. What is the volumetric density of electric charge and current density?

MATLAB Application Example

Task.

Given: Circuit with electric current I in space represents the perimeter of a triangle, the Cartesian coordinates of the vertices of which are given: x 1 , x 2 , x 3 , y 1 , y 2 , y 3 , z 1 , z 2 , z 3. Here the subscripts are the numbers of the vertices. The vertices are numbered in the direction of flow of electric current.

Required compose a MATLAB function that calculates the dipole magnetic moment vector of the loop. When compiling an m-file, it can be assumed that spatial coordinates are measured in meters, and current in amperes. Arbitrary organization of input and output parameters is allowed.

Solution

% m_dip_moment - calculation of the magnetic dipole moment of a triangular circuit with a current in space

% pm = m_dip_moment(tok,nodes)

% INPUT PARAMETERS

% tok - current in the circuit;

% nodes is a square matrix of the form ".", each row of which contains the coordinates of the corresponding vertex.

% OUTPUT PARAMETER

% pm is a row matrix of the Cartesian components of the magnetic dipole moment vector.

function pm = m_dip_moment(tok,nodes);

pm=tok*)]) det()]) det()])]/2;

% In the last statement, the triangle area vector is multiplied by the current

>> nodes=10*rand(3)

9.5013 4.8598 4.5647

2.3114 8.913 0.18504

6.0684 7.621 8.2141

>> pm=m_dip_moment(1,nodes)

13.442 20.637 -2.9692

In this case it worked P M = (13.442* 1 x + 20.637*1 y - 2.9692*1 z) A*m 2 if the current in the circuit is 1 A.

§ 1.4. Spatial differential operators in electromagnetic field theory

Gradient scalar field Φ( Q) = Φ( x, y, z) is a vector field defined by the formula:

,

Where V 1 - area containing the point Q; S 1 - closed surface bounding the area V 1 , Q 1 - point belonging to the surface S 1 ; δ - greatest distance from the point Q to points on the surface S 1 (max| Q Q 1 |).

Divergence vector field F (Q)=F (x, y, z) is called a scalar field, defined by the formula:

Rotor(vortex) vector field F (Q)=F (x, y, z) is a vector field defined by the formula:

rot F =

Nabla operator is a vector differential operator, which in Cartesian coordinates is defined by the formula:

Let's represent grad, div and rot through the nabla operator:

Let's write these operators in Cartesian coordinates:

; ;

The Laplace operator in Cartesian coordinates is defined by the formula:

Second order differential operators:

Integral theorems

Gradient theorem ;

Divergence theorem

Rotor theorem

In the theory of EMF, one more of the integral theorems is also used:

.

Control questions

1. What is called the scalar field gradient?

2. What is called the divergence of a vector field?

3. What is called the curl of a vector field?

4. What is the nabla operator and how are first-order differential operators expressed through it?

5. What integral theorems are true for scalar and vector fields?

MATLAB Application Example

Task.

Given: In the volume of a tetrahedron, the scalar and vector fields change according to a linear law. The coordinates of the tetrahedron vertices are specified by a matrix of the form [ x 1 , y 1 , z 1 ; x 2 , y 2 , z 2 ; x 3 , y 3 , z 3 ; x 4 , y 4 , z 4 ]. The values ​​of the scalar field at the vertices are specified by the matrix [Ф 1 ; F 2; F 3; F 4]. The Cartesian components of the vector field at the vertices are specified by the matrix [ F 1 x, F 1y, F 1z; F 2x, F 2y, F 2z; F 3x, F 3y, F 3z; F 4x, F 4y, F 4z].

Define in the volume of the tetrahedron, the gradient of the scalar field, as well as the divergence and curl of the vector field. Write a MATLAB function for this.

Solution. Below is the text of the m-function.

% grad_div_rot - Calculate gradient, divergence and rotor... in the volume of a tetrahedron

% =grad_div_rot(nodes,scalar,vector)

% INPUT PARAMETERS

% nodes - matrix of coordinates of tetrahedron vertices:

% rows correspond to vertices, columns - coordinates;

% scalar - columnar matrix of scalar field values ​​at the vertices;

% vector - matrix of vector field components at vertices:

% OUTPUT PARAMETERS

% grad - row matrix of Cartesian components of the gradient of the scalar field;

% div - the divergence value of the vector field in the volume of the tetrahedron;

% rot is a row matrix of the Cartesian components of the vector field rotor.

% In the calculations it is assumed that in the volume of the tetrahedron

% vector and scalar fields vary in space according to a linear law.

function =grad_div_rot(nodes,scalar,vector);

a=inv(); % Linear interpolation coefficient matrix

grad=(a(2:end,:)*scalar)."; % Gradient components of the scalar field

div=*vector(:); % Vector field divergence

rot=sum(cross(a(2:end,:),vector."),2).";

An example of running the developed m-function:

>> nodes=10*rand(4,3)

3.5287 2.0277 1.9881

8.1317 1.9872 0.15274

0.098613 6.0379 7.4679

1.3889 2.7219 4.451

>> scalar=rand(4,1)

>> vector=rand(4,3)

0.52515 0.01964 0.50281

0.20265 0.68128 0.70947

0.67214 0.37948 0.42889

0.83812 0.8318 0.30462

>> =grad_div_rot(nodes,scalar,vector)

0.16983 -0.03922 -0.17125

0.91808 0.20057 0.78844

If we assume that spatial coordinates are measured in meters, and vector and scalar fields are dimensionless, then in this example we get:

grad Ф = (-0.16983* 1 x - 0.03922*1 y - 0.17125*1 z) m -1 ;

div F = -1.0112 m -1 ;

rot F = (-0.91808*1 x + 0.20057*1 y + 0.78844*1 z) m -1 .

§ 1.5. Basic laws of electromagnetic field theory

EMF equations in integral form

Total current law:

or

Circulation of the magnetic field strength vector along the contour l equal to the total electric current flowing through the surface S, stretched on the contour l, if the direction of the current forms a right-handed system with the direction of bypassing the circuit.

Law of electromagnetic induction:

,

Where E c is the intensity of the external electric field.

EMF electromagnetic induction e and in the circuit l equal to the rate of change of magnetic flux through the surface S, stretched on the contour l, and the direction of the rate of change of magnetic flux forms with the direction e and a left-handed screw system.

Gauss's theorem in integral form:

Electric displacement vector flow through a closed surface S equal to the sum of free electric charges in the volume limited by the surface S.

Law of continuity of magnetic induction lines:

The magnetic flux through any closed surface is zero.

Direct application of equations in integral form makes it possible to calculate the simplest electromagnetic fields. To calculate electromagnetic fields of more complex shapes, equations in differential form are used. These equations are called Maxwell's equations.

Maxwell's equations for stationary media

These equations follow directly from the corresponding equations in integral form and from the mathematical definitions of spatial differential operators.

Total current law in differential form:

,

Total electric current density,

Density of external electric current,

Conduction current density,

Bias current density: ,

Transfer current density: .

This means that the electric current is a vortex source of the vector field of magnetic field strength.

The law of electromagnetic induction in differential form:

This means that the alternating magnetic field is a vortex source for the spatial distribution of the electric field strength vector.

Equation of continuity of magnetic induction lines:

This means that the field of the magnetic induction vector has no sources, i.e. There are no magnetic charges (magnetic monopoles) in nature.

Gauss's theorem in differential form:

This means that the sources of the vector field of electric displacement are electric charges.

To ensure the uniqueness of the solution to the problem of EMF analysis, it is necessary to supplement Maxwell’s equations with equations of material connections between vectors E And D , and B And H .

Relationships between field vectors and electrical properties of the medium

It is known that

All dielectrics are polarized under the influence of an electric field. All magnets are magnetized under the influence of a magnetic field. The static dielectric properties of a substance can be completely described by the functional dependence of the polarization vector P from the electric field strength vector E (P =P (E )). The static magnetic properties of a substance can be completely described by the functional dependence of the magnetization vector M from the magnetic field strength vector H (M =M (H )). In the general case, such dependences are ambiguous (hysteretic) in nature. This means that the polarization or magnetization vector at a point Q is determined not only by the value of the vector E or H at this point, but also the background of the change in vector E or H at this point. It is extremely difficult to experimentally study and model these dependencies. Therefore, in practice it is often assumed that the vectors P And E , and M And H are collinear, and the electrical properties of a substance are described by scalar hysteresis functions (| P |=|P |(|E |), |M |=|M |(|H |). If the hysteresis characteristics of the above functions can be neglected, then the electrical properties are described by unambiguous functions P=P(E), M=M(H).

In many cases, these functions can be approximately considered linear, i.e.

Then, taking into account relation (1), we can write the following

Accordingly, the relative dielectric and magnetic permeability of the substance:

Absolute dielectric constant of a substance:

Absolute magnetic permeability of a substance:

Relations (2), (3), (4) characterize the dielectric and magnetic properties of the substance. The electrically conductive properties of a substance can be described by Ohm's law in differential form

where is the specific electrical conductivity of the substance, measured in S/m.

In a more general case, the relationship between the conduction current density and the electric field strength vector has a nonlinear vector-hysteresis character.

Electromagnetic field energy

The volumetric energy density of the electric field is equal to

,

Where W e is measured in J/m 3.

The volumetric energy density of the magnetic field is equal to

,

Where W m is measured in J/m 3.

The volumetric energy density of the electromagnetic field is equal to

In the case of linear electrical and magnetic properties of matter, the volumetric energy density of the EMF is equal to

This expression is valid for instantaneous values ​​of specific energy and EMF vectors.

Specific power of heat losses from conduction currents

Power density of third party sources

Control questions

1. How is the law of total current formulated in integral form?

2. How is the law of electromagnetic induction formulated in integral form?

3. How are Gauss’s theorem and the law of magnetic flux continuity formulated in integral form?

4. How is the total current law formulated in differential form?

5. How is the law of electromagnetic induction formulated in differential form?

6. How are Gauss’s theorem and the law of continuity of magnetic induction lines formulated in integral form?

7. What relationships describe the electrical properties of a substance?

8. How is the energy of the electromagnetic field expressed through the vector quantities that determine it?

9. How is the specific power of heat losses and the specific power of third-party sources determined?

MATLAB Application Examples

Problem 1.

Given: Inside the volume of the tetrahedron, the magnetic induction and magnetization of the substance change according to a linear law. The coordinates of the vertices of the tetrahedron are given, the values ​​of the vectors of magnetic induction and magnetization of the substance at the vertices are also given.

Calculate electric current density in the volume of the tetrahedron, using the m-function compiled when solving the problem in the previous paragraph. Perform the calculation in the MATLAB command window, assuming that spatial coordinates are measured in millimeters, magnetic induction in tesla, magnetic field strength and magnetization in kA/m.

Solution.

Let's set the initial data in a format compatible with the m-function grad_div_rot:

>> nodes=5*rand(4,3)

0.94827 2.7084 4.3001

0.96716 0.75436 4.2683

3.4111 3.4895 2.9678

1.5138 1.8919 2.4828

>> B=rand(4.3)*2.6-1.3

1.0394 0.41659 0.088605

0.83624 -0.41088 0.59049

0.37677 -0.54671 -0.49585

0.82673 -0.4129 0.88009

>> mu0=4e-4*pi % absolute magnetic permeability of vacuum, µH/mm

>> M=rand(4,3)*1800-900

122.53 -99.216 822.32

233.26 350.22 40.663

364.93 218.36 684.26

83.828 530.68 -588.68

>> =grad_div_rot(nodes,ones(4,1),B/mu0-M)

0 -3.0358e-017 0

914.2 527.76 -340.67

In this example, the vector of the total current density in the volume under consideration turned out to be equal to (-914.2* 1 x + 527.76*1 y - 340.67*1 z) A/mm 2 . To determine the modulus of the current density, we execute the following operator:

>> cur_d=sqrt(cur_dens*cur_dens.")

The calculated value of current density cannot be obtained in highly magnetized environments in real technical devices. This example is purely educational. Now let’s check the correctness of specifying the distribution of magnetic induction in the volume of the tetrahedron. To do this, we execute the following statement:

>> =grad_div_rot(nodes,ones(4,1),B)

0 -3.0358e-017 0

0.38115 0.37114 -0.55567

Here we got the div value B = -0.34415 T/mm, which cannot be in accordance with the law of continuity of magnetic induction lines in differential form. It follows from this that the distribution of magnetic induction in the volume of the tetrahedron is specified incorrectly.

Problem 2.

Let a tetrahedron, the coordinates of the vertices of which are given, be in the air (units of measurement are meters). Let the values ​​of the electric field strength vector at its vertices be given (units of measurement - kV/m).

Required calculate the volumetric charge density inside the tetrahedron.

Solution can be done similarly:

>> nodes=3*rand(4,3)

2.9392 2.2119 0.59741

0.81434 0.40956 0.89617

0.75699 0.03527 1.9843

2.6272 2.6817 0.85323

>> eps0=8.854e-3% absolute dielectric constant of vacuum, nF/m

>> E=20*rand(4,3)

9.3845 8.4699 4.519

1.2956 10.31 11.596

19.767 6.679 15.207

11.656 8.6581 10.596

>> =grad_div_rot(nodes,ones(4,1),E*eps0)

0.076467 0.21709 -0.015323

In this example, the volumetric charge density was equal to 0.10685 µC/m 3.

§ 1.6. Boundary conditions for EMF vectors.
Law of conservation of charge. Umov-Poynting theorem

or

Here it is indicated: H 1 - vector of magnetic field strength at the interface between media in medium No. 1; H 2 - the same in environment No. 2; H 1t- tangential (tangent) component of the magnetic field strength vector at the interface between media in medium No. 1; H 2t- the same in environment No. 2; E 1 vector of the total electric field strength at the interface between media in medium No. 1; E 2 - the same in environment No. 2; E 1 c - third-party component of the electric field strength vector at the interface between media in medium No. 1; E 2c - the same in environment No. 2; E 1t- tangential component of the electric field strength vector at the interface between media in medium No. 1; E 2t- the same in environment No. 2; E 1s t- tangential third-party component of the electric field strength vector at the interface between media in medium No. 1; E 2t- the same in environment No. 2; B 1 - vector of magnetic induction at the interface between media in medium No. 1; B 2 - the same in environment No. 2; B 1n- normal component of the magnetic induction vector at the interface between media in medium No. 1; B 2n- the same in environment No. 2; D 1 - electric displacement vector at the interface between media in medium No. 1; D 2 - the same in environment No. 2; D 1n- normal component of the electric displacement vector at the interface between media in medium No. 1; D 2n- the same in environment No. 2; σ is the surface density of the electric charge at the interface, measured in C/m2.

Law of conservation of charge

If there are no third-party current sources, then

,

and in the general case, i.e., the total current density vector has no sources, i.e., the total current lines are always closed

The volumetric power density consumed by a material point in an EMF is equal to

In accordance with identity (1)

This is the power balance equation for volume V. In the general case, in accordance with equality (3), the electromagnetic power generated by sources inside the volume V, goes to heat losses, to the accumulation of EMF energy and to radiation into the surrounding space through a closed surface that limits this volume.

The integrand in integral (2) is called the Poynting vector:

,

Where P measured in W/m2.

This vector is equal to the electromagnetic power flux density at some observation point. Equality (3) is a mathematical expression of the Umov-Poynting theorem.

Electromagnetic power emitted by the area V into the surrounding space is equal to the flux of the Poynting vector through a closed surface S, limiting the area V.

Control questions

1. What expressions describe the boundary conditions for the electromagnetic field vectors at the interfaces between media?

2. How is the law of conservation of charge formulated in differential form?

3. How is the law of conservation of charge formulated in integral form?

4. What expressions describe the boundary conditions for the current density at the interfaces?

5. What is the volumetric power density consumed by a material point in an electromagnetic field?

6. How is the electromagnetic power balance equation written for a certain volume?

7. What is a Poynting vector?

8. How is the Umov-Poynting theorem formulated?

MATLAB Application Example

Task.

Given: There is a triangular surface in space. The coordinates of the vertices are given. The values ​​of the electric and magnetic field strength vectors at the vertices are also specified. The third-party component of the electric field strength is zero.

Required calculate the electromagnetic power passing through this triangular surface. Write a MATLAB function that performs this calculation. When calculating, assume that the positive normal vector is directed in such a way that if viewed from its end, the movement in increasing order of vertex numbers will occur counterclockwise.

Solution. Below is the text of the m-function.

% em_power_tri - calculation of electromagnetic power passing through

% triangular surface in space

% P=em_power_tri(nodes,E,H)

% INPUT PARAMETERS

% nodes is a square matrix of the form ",

% in each line of which the coordinates of the corresponding vertex are written.

% E - matrix of components of the electric field strength vector at the vertices:

% rows correspond to vertices, columns - Cartesian components.

% H - matrix of components of the magnetic field strength vector at the vertices.

% OUTPUT PARAMETER

% P - electromagnetic power passing through the triangle

% During calculations it is assumed that on the triangle

% field strength vectors change in space according to a linear law.

function P=em_power_tri(nodes,E,H);

% Calculate the double area vector of the triangle

S=)]) det()]) det()])];

P=sum(cross(E,(ones(3,3)+eye(3))*H,2))*S."/24;

An example of running the developed m-function:

>> nodes=2*rand(3,3)

0.90151 0.5462 0.4647

1.4318 0.50954 1.6097

1.7857 1.7312 1.8168

>> E=2*rand(3,3)

0.46379 0.15677 1.6877

0.47863 1.2816 0.3478

0.099509 0.38177 0.34159

>>H=2*rand(3,3)

1.9886 0.62843 1.1831

0.87958 0.73016 0.23949

0.6801 0.78648 0.076258

>> P=em_power_tri(nodes,E,H)

If we assume that spatial coordinates are measured in meters, the electric field strength vector is in volts per meter, and the magnetic field strength vector is in amperes per meter, then in this example the electromagnetic power passing through the triangle is equal to 0.18221 W.

In this lesson, the topic of which is “Electromagnetic field,” we will discuss the concept of “electromagnetic field,” the features of its manifestation and the parameters of this field.

We are talking on a mobile phone. How is the signal transmitted? How is the signal transmitted from a space station flying to Mars? In the void? Yes, there may be no substance, but this is not emptiness, there is something else through which the signal is transmitted. This something was called an electromagnetic field. This is not a directly observable, but a really existing object of nature.

If a sound signal is a change in the parameters of a substance, for example air (Fig. 1), then a radio signal is a change in the parameters of the EM field.

Rice. 1. Sound wave propagation in air

The words “electric” and “magnetic” are clear to us, we have already studied separately electrical phenomena (Fig. 2) and magnetic phenomena (Fig. 3), but why then are we talking about the electromagnetic field? Today we will figure it out.

Rice. 2. Electric field

Rice. 3. Magnetic field

Examples of electromagnetic phenomena.

A microwave creates strong, and most importantly, very rapidly changing electromagnetic fields that act on an electric charge. And as we know, atoms and molecules of substances contain an electric charge (Fig. 4). This is where the electromagnetic field acts on it, forcing the molecules to move faster (Fig. 5) - the temperature increases and the food heats up. X-rays, ultraviolet rays, and visible light have the same nature.

Rice. 4. The water molecule is a dipole

Rice. 5. Movement of molecules having an electrical charge

In a microwave oven, the electromagnetic field imparts energy to the substance, which is used for heating, visible light imparts energy to the eye receptors, which is used to activate the receptor (Fig. 6), the energy of ultraviolet rays is used to form melanin in the skin (the appearance of tanning, Fig. 7), and The energy of X-rays causes the film to turn black, on which you can see an image of your skeleton (Fig. 8). The electromagnetic field in all these cases has different parameters, and therefore has different effects.

Rice. 6. Conditional diagram of activation of the eye receptor by visible light energy

Rice. 7. Skin tanning

Rice. 8. Blackening of the film during x-ray

So we encounter the electromagnetic field much more often than it seems, and have long been accustomed to the phenomena that are associated with it.

So, we know that the electric field arises around electric charges (Fig. 9). Everything is clear here.

Rice. 9. Electric field around an electric charge

If an electric charge moves, then, as we studied, a magnetic field arises around it (Fig. 10). Here the question already arises: an electric charge is moving, there is an electric field around it, what does the magnetic field have to do with it? One more question: we say “the charge is moving.” But motion is relative, and it can move in one frame of reference and be at rest in another (Fig. 11). Does this mean that a magnetic field will exist in one frame of reference, but not in another? But the field should not exist or not exist depending on the choice of reference frame.

Rice. 10. Magnetic field around a moving electric charge

Rice. 11. Relativity of charge motion

The fact is that there is a single electromagnetic field, and it has a single source - an electric charge. It has two components. Electric and magnetic fields are separate manifestations, separate components of a single electromagnetic field, which manifest themselves differently in different reference systems (Fig. 12).

Rice. 12. Manifestations of the electromagnetic field

You can choose a reference frame in which only the electric field, or only the magnetic field, or both at once will appear. However, it is impossible to choose a reference system in which both the electric and magnetic components will be zero, that is, in which the electromagnetic field will cease to exist.

Depending on the reference system, we see either one component of the field, or another, or both. It’s like the movement of a body in a circle: if you look at such a body from above, we will see movement along the circle (Fig. 13), if from the side, we will see oscillations along the segment (Fig. 14). In each projection onto the coordinate axis, circular motion is oscillations.

Rice. 13. Body movement in a circle

Rice. 14. Body oscillations along a segment

Rice. 15. Projection of circular movements onto the coordinate axis

Another analogy is the projection of a pyramid onto a plane. It can be projected into a triangle or a square. On the plane these are completely different figures, but all of this is a pyramid, which is looked at from different sides. But there is no angle from which the pyramid will disappear completely. It will just look more like a square or triangle (Fig. 16).

Rice. 16. Projections of a pyramid onto a plane

Consider a conductor carrying current. In it, negative charges are compensated by positive ones, the electric field around it is zero (Fig. 17). The magnetic field is not zero (Fig. 18); we considered the emergence of a magnetic field around a conductor with current. Let us choose a reference system in which the electrons forming the electric current will be stationary. But in this reference frame, the positively charged ions of the conductor will move in the opposite direction relative to the electrons: a magnetic field still arises (Fig. 18).

Rice. 17. A conductor with current whose electric field is zero

Rice. 18. Magnetic field around a current-carrying conductor

If electrons were in a vacuum, in this reference frame an electric field would arise around them, because they are not compensated by positive charges, but there would be no magnetic field (Fig. 19).

Rice. 19. Electric field around electrons in a vacuum

Let's look at another example. Let's take a permanent magnet. There is a magnetic field around it, but no electric one. Indeed, the electric field of protons and electrons is compensated (Fig. 20).

Rice. 20. Magnetic field around a permanent magnet

Let us take a reference frame in which the magnet is moving. A vortex electric field will appear around a moving permanent magnet (Fig. 21). How to identify it? Let us place a metal ring (immobile in this reference frame) in the path of the magnet. A current will arise in it - this is a well-known phenomenon of electromagnetic induction: when the magnetic flux changes, an electric field arises, leading to the movement of charges, to the appearance of a current (Fig. 22). In one reference frame there is no electric field, but in another it appears.

Rice. 21. Vortex electric field around a moving permanent magnet

Rice. 22. The phenomenon of electromagnetic induction

Magnetic field of a permanent magnet

In any substance, the electrons that revolve around the nucleus can be thought of as a small electric current that flows in a circle (Fig. 23). This means that a magnetic field arises around it. If the substance is not magnetic, it means that the planes of rotation of the electrons are directed arbitrarily and the magnetic fields from individual electrons compensate each other, since they are directed chaotically.

Rice. 23. Representation of the rotation of electrons around the nucleus

In magnetic substances, the planes of electron rotation are oriented approximately equally (Fig. 24). Therefore, the magnetic fields from all electrons add up, and a non-zero magnetic field is obtained on the scale of the entire magnet.

Rice. 24. Rotation of electrons in magnetic substances

There is a magnetic field around a permanent magnet, or rather the magnetic component of the electromagnetic field (Fig. 25). Can we find a frame of reference in which the magnetic component becomes zero and the magnet loses its properties? Still no. Indeed, electrons rotate in the same plane (see Fig. 24); at any moment in time, the speeds of the electrons are not directed in the same direction (Fig. 26). So it is impossible to find a frame of reference where they all freeze and the magnetic field disappears.

Rice. 25. Magnetic field around a permanent magnet

Thus, electric and magnetic fields are different manifestations of a single electromagnetic field. It cannot be said that at a specific point in space there is only a magnetic or only an electric field. There may be one or the other. It all depends on the frame of reference from which we view this point.

Why did we previously talk separately about electric and magnetic fields? Firstly, it happened historically: people have known about magnets for a long time, people have long observed fur electrified on amber, and no one realized that these phenomena were of the same nature. And secondly, this is a convenient model. In problems where we are not interested in the relationship between the electric and magnetic components, it is convenient to consider them separately. Two charges at rest in a given reference frame interact through an electric field - we apply Coulomb’s law to them, we are not interested in the fact that these same electrons can move in some reference frame and create a magnetic field, and we successfully solve the problem (Fig. 27) .

Rice. 27. Coulomb's Law

The effect of a magnetic field on a moving charge is considered in another model, and within the framework of its applicability, it also works perfectly in solving a number of problems (Fig. 28).

Rice. 28. Left hand rule

Let's try to understand how the components of the electromagnetic field are interconnected.

It is worth noting that the exact relationship is quite complex. It was developed by British physicist James Maxwell. He derived the famous 4 Maxwell equations (Fig. 29), which are studied in universities and require knowledge of higher mathematics. We will not study them, of course, but in a few simple words we will understand what they mean.

Rice. 29. Maxwell's equations

Maxwell relied on the work of another physicist - Faraday (Fig. 30), who simply qualitatively described all the phenomena. He made drawings (Fig. 31) and notes that greatly helped Maxwell.

Rice. 31. Drawings by Michael Faraday from the book “Electricity” (1852)

Faraday discovered the phenomenon of electromagnetic induction (Fig. 32). Let's remember what it is. An alternating magnetic field generates an induced emf in a conductor. In other words, an alternating magnetic field (yes, in this case, not an electric charge) generates an electric field. This electric field is vortex, that is, its lines are closed (Fig. 33).

Rice. 32. Drawings by Michael Faraday for the experiment

Rice. 33. Occurrence of induced emf in a conductor

In addition, we know that a magnetic field is generated by a moving electric charge. It would be more correct to say that it is generated by an alternating electric field. As the charge moves, the electric field at each point changes, and this change generates a magnetic field (Fig. 34).

Rice. 34. Emergence of a magnetic field

You can notice the appearance of a magnetic field between the plates of the capacitor. When it charges or discharges, an alternating electric field is generated between the plates, which in turn generates a magnetic field. In this case, the magnetic field lines will lie in a plane perpendicular to the electric field lines (Fig. 35).

Rice. 35. The appearance of a magnetic field between the capacitor plates

Now let's look at Maxwell's equations (Fig. 29), a short explanation of them is given below for your reference.

The divergence icon is a mathematical operator; it highlights that component of the field that has a source, that is, the field lines begin and end at something. Look at the second equation: this component of the magnetic field is zero: magnetic field lines do not start or end at anything, there is no magnetic charge. Look at the first equation: this component of the electric field is proportional to the charge density. An electric field is created by an electric charge.

The most interesting are the following two equations. The rotor icon is a mathematical operator that highlights the vortex component of the field. The third equation means that the vortex electric field is created by a time-varying magnetic field (this is the derivative, which, as you know from mathematics, means the rate of change of the magnetic field). That is, we are talking about electromagnetic induction.

The fourth equation shows, if you do not pay attention to the proportionality coefficients: the vortex magnetic field is created by a changing electric field, as well as by an electric current ( - current density). We are talking about what we know well: a magnetic field is created by a moving electric charge and.

As you can see, an alternating magnetic field can generate an alternating electric field, and an alternating electric field, in turn, generates an alternating magnetic field, and so on (Fig. 36).

Rice. 36. An alternating magnetic field can generate an alternating electric field, and vice versa

As a result, an electromagnetic wave can be formed in space (Fig. 37). These waves have different manifestations - these are radio waves, visible light, ultraviolet and so on. We'll talk about this in the next lessons.

Rice. 37. Electromagnetic wave

Bibliography

1. Kasyanov V.A. Physics. 11th grade: Educational. for general education institutions. - M.: Bustard, 2005.
2. Myakishev G.Ya. Physics: Textbook. for 11th grade general education institutions. - M.: Education, 2010.

1. Internet portal “studopedia.su” ()
2. Internet portal “worldofschool.ru” ()

Homework

1. Is it possible to detect a magnetic field in a reference frame associated with one of the uniformly moving electrons in the flow that is created in the TV picture tube?
2. What field appears around an electron moving in a given frame of reference with constant speed?
3. What kind of field can be detected around motionless amber charged with static electricity? Around a moving one? Justify your answers.