Fundamentals of electrodynamics. Field equation in a dielectric

The subject of classical electrodynamics

Classical electrodynamics is a theory that explains the behavior of an electromagnetic field that carries out electromagnetic interaction between electric charges.

The laws of classical macroscopic electrodynamics are formulated in Maxwell's equations, which allow you to determine the values ​​of the characteristics of the electromagnetic field: the electric field strength E and magnetic induction AT in vacuum and in macroscopic bodies, depending on the distribution of electric charges and currents in space.

The interaction of stationary electric charges is described by the equations of electrostatics, which can be obtained as a consequence of Maxwell's equations.

The microscopic electromagnetic field created by individual charged particles in classical electrodynamics is determined by the Lorentz-Maxwell equations, which underlie the classical statistical theory of electromagnetic processes in macroscopic bodies. Averaging these equations leads to Maxwell's equations.

Among all known types of interaction, electromagnetic interaction ranks first in terms of breadth and variety of manifestations. This is due to the fact that all bodies are built of electrically charged (positive and negative) particles, the electromagnetic interaction between which, on the one hand, is many orders of magnitude more intense than the gravitational and weak one, and on the other hand, is long-range, in contrast to the strong interaction.

Electromagnetic interaction determines the structure of atomic shells, the adhesion of atoms into molecules (chemical bond forces) and the formation of condensed matter (interatomic interaction, intermolecular interaction).

The laws of classical electrodynamics are inapplicable at high frequencies and, accordingly, small lengths of electromagnetic waves, i.e. for processes occurring on small space-time intervals. In this case, the laws of quantum electrodynamics are valid.

1.2. Electric charge and its discreteness.
Short range theory

The development of physics has shown that the physical and chemical properties of a substance are largely determined by the forces of interaction due to the presence and interaction of electric charges of molecules and atoms of various substances.

It is known that in nature there are two types of electric charges: positive and negative. They can exist in the form of elementary particles: electrons, protons, positrons, positive and negative ions, etc., as well as "free electricity", but only in the form of electrons. Therefore, a positively charged body is a collection of electric charges with a lack of electrons, and a negatively charged body - with their excess. Charges of different signs compensate each other, therefore, in uncharged bodies there are always charges of both signs in such quantities that their total effect is compensated.

redistribution process positive and negative charges of uncharged bodies, or among separate parts of the same body, under the influence various factors called electrification.

Since the redistribution of free electrons occurs during electrification, for example, both interacting bodies are electrified, one of them being positive and the other negative. The number of charges (positive and negative) remains unchanged.

This implies the conclusion that charges are not created and do not disappear, but only redistributed between interacting bodies and parts of the same body, quantitatively remaining unchanged.

This is the meaning of the law of conservation of electric charges, which can be written mathematically as follows:

those. in an isolated system, the algebraic sum of electric charges remains constant.

An isolated system is understood as a system through which no other substance penetrates, with the exception of photons of light, neutrons, since they do not carry a charge.

It must be borne in mind that the total electric charge of an isolated system is relativistically invariant, since observers located in any given inertial coordinate system, measuring the charge, get the same value.

A number of experiments, in particular the laws of electrolysis, Millikan's experiment with a drop of oil, have shown that in nature electric charges are discrete to the charge of an electron. Any charge is a multiple of an integer number of the electron charge.

In the process of electrification, the charge changes discretely (quantized) by the value of the electron charge. Charge quantization is a universal law of nature.

In electrostatics, the properties and interactions of charges that are immobile in the frame of reference in which they are located are studied.

The presence of an electric charge in bodies causes them to interact with other charged bodies. At the same time, bodies charged with the same name repel each other, and charged oppositely, they attract.

The theory of short-range interaction is one of the theories of interaction in physics. In physics, interaction is understood as any influence of bodies or particles on each other, leading to a change in the state of their motion.

In Newtonian mechanics, the mutual action of bodies on each other is quantitatively characterized by force. More common characteristic interaction is potential energy.

Initially, in physics, the idea was established that the interaction between bodies can be carried out directly through empty space, which does not take part in the transfer of interaction. The transfer of interaction occurs instantly. Thus, it was believed that the movement of the Earth should immediately lead to a change in the gravitational force acting on the Moon. This was the meaning of the so-called theory of interaction, called the theory of long-range action. However, these ideas were abandoned as untrue after the discovery and study of the electromagnetic field.

It was proved that the interaction of electrically charged bodies is not instantaneous and the movement of one charged particle leads to a change in the forces acting on other particles, not at the same moment, but only after a finite time.

Each electrically charged particle creates an electromagnetic field that acts on other particles, i.e. interaction is transmitted through an "intermediary" - an electromagnetic field. The speed of propagation of an electromagnetic field is equal to the speed of propagation of light in a vacuum. A new theory of interaction arose - the theory of short-range interaction.

According to this theory, the interaction between bodies is carried out through certain fields (for example, gravitation through a gravitational field), continuously distributed in space.

After the advent of quantum field theory, the concept of interactions has changed significantly.

According to quantum theory, any field is not continuous, but has a discrete structure.

Owing to corpuscular-wave dualism, certain particles correspond to each field. Charged particles continuously emit and absorb photons, which form the electromagnetic field surrounding them. The electromagnetic interaction in quantum field theory is the result of the exchange of particles by photons (quanta) of the electromagnetic field, i.e. photons are carriers of such interaction. Similarly, other types of interactions arise as a result of the exchange of particles by quanta of the corresponding fields.

Despite the variety of influences of bodies on each other (depending on the interaction of their constituent elementary particles), in nature, according to modern data, there are only four types of fundamental interactions: gravitational, weak, electromagnetic and strong (in order of increasing interaction intensity). The intensities of interactions are determined by coupling constants (in particular, the electric charge for electromagnetic interaction is a coupling constant).

Modern quantum theory electromagnetic interaction perfectly describes all known electromagnetic phenomena.

In the 60-70s of the century, a unified theory of weak and electromagnetic interactions (the so-called electroweak interaction) of leptons and quarks was basically built.

The modern theory of the strong interaction is quantum chromodynamics.

Attempts are being made to combine the electroweak and strong interactions into the so-called "Great Unification", as well as to include them in a single scheme of gravitational interaction.

• Electrodynamics studies electromagnetic processes in vacuum and in matter - in dielectrics, magnets, conductors, semiconductors, superconductors, electrolytes and plasma.

• Classical electrodynamics studies a continuous non-quantized classical electromagnetic field and electromagnetic processes in this field associated with charges and currents, as well as the relativism of these processes.

• The fundamental laws of classical electrodynamics are Maxwell's equations and material equations.

1.1 Electric charge. Electric moment

1.1.1 Electric charge

• Electric charge is a fundamental property of matter. Charge does not exist separately from matter. Charge carriers are elementary particles and material bodies.

1.1.2 Elementary charge

• The elementary charge is the smallest positive or negative charge equal in magnitude to the charge of an electron.

1.1.3 Macroscopic charge

• The carrier of a macroscopic charge is a material body. A charge consists of an integer number of elementary charges

Integer

1.1.4 Law of conservation of charge

• When redistributing the charge between objects closed system total charge is conserved

1.1.5 Charge distributed over the volume of the body

is the bulk charge density,

is the charge of the volume element,

is the volume charge of the whole body.

1.1.6 Charge distributed over the surface of the body

• — surface density charge,

is the charge of the surface element,

is the surface charge of the whole body.

1.1.7 Charge distributed over a linear body

• is the linear charge density,

is the charge of the length element,

is the linear charge of the whole body.

1.1.8 Polar systems of bound charges

• In a polar system, charges of opposite signs are separated, and the system itself is electrically neutral. Variants of such systems: dipole, quadrupole, octupole, ..., multipole. The carriers of polar charges can be particles of matter - atoms, molecules, elements of the crystal lattice, as well as macroscopic bodies. Main characteristic polar system is its electric moment. This is a vector quantity through which the interaction of the polar system with the electric field is expressed.

1.1.9 Electric moment of the dipole

• A dipole is a two-pole system. These are two charges, equal and opposite in sign, separated from one another by a distance. The electric moment of the dipole is the vector

directed along the dipole axis from the negative to the positive pole.

1.2 Magnetic charge. Magnetic moment

1.2.1 Magnetic monopole

• This particle is a carrier of a positive or negative elementary magnetic charge. The existence of such a particle was justified theoretically by Dirac in 1931, but it has not yet been experimentally discovered.

1.2.2 Magnetic moment of matter particles

Electrons, atoms, molecules and other particles of matter have a magnetic moment. This is the main magnetic characteristic of particles, which determines their interaction with a magnetic field. In contrast to the magnetic charge, the magnetic moment is reliably confirmed by experiment and is considered as primary information about the magnetic properties of particles.

1.2.3 Coulomb model of the magnetic moment

• The real magnetic moment of a particle can be formally associated as a model with the moment of an imaginary magnetic dipole

where is the magnetic charge of the poles, and is the vector distance between the poles. Although there are no magnetic charges, this model of the magnetic moment, introduced into physics in the past, turned out to be formally convenient and in many cases is used now, as having only a virtual meaning in intermediate calculations.

1.2.4 Ampère model of the magnetic moment

• The real magnetic moment of a particle can also be formally associated as a model with the magnetic moment of an imaginary flat coil with current

where is the current in the coil, is the vector area of ​​the coil, is the unit normal to the surface, related to the direction of the current by the rule of the right screw. In this model, it is assumed that the current loop covers the particle and the current, called molecular current, flows around it. This current should be considered as a formal one, which means that the Ampere model of the magnetic moment is just as virtual as the Coulomb one, although it surpasses the latter in many theoretical calculations.

1.2.5 Comparison of magnetic moment models

• A magnetic dipole is similar to an electric one and their moments are determined by similar expressions. A coil with current is not similar to a magnetic dipole, but they are fully similar and equivalent to each other in magnetic moment and in their interaction with the magnetic field (Fig. 1.2.5). The choice of the Coulomb or Ampere model of the magnetic moment is determined by which of them leads to more deep understanding and calculation of the magnetic state of matter.

Fig 1.2.5

Electric and magnesium t moment of matter particles

1.3 Electric and magnetic polarization of matter

1.3.1 Orienting action of the electric field on a particle of matter

• If a particle of matter has an electric moment, then the electric field strength turns the particle, while the torque

Under the action, the electric moment is oriented in the direction of the field (Fig. 1.3.1).

Rice. 1.3.1

Orienting action t fields and on particles with an electric momentum n volume or magnetic m about the cop

1.3.2 Orienting action of a magnetic field on a particle of matter

• If a particle of a substance has a magnetic moment, then the magnetic field with induction turns the particle, while the torque

Under the action, the magnetic moment is oriented in the direction of the magnetic field, and the plane of the coil with current is set perpendicular to the field (Fig. 1.3.1).

1.3.3 Electric polarization of matter (dielectric)

• An external electric field produces a massive orienting effect on the electric moments of all particles and brings the substance into a state of electric polarization. The degree of polarization of a substance is characterized by the polarization vector.

This value is local, since the total electric moment does not refer to the entire substance, but to its elementary part by volume. With uniform polarization, the vector has the same value at all points of the substance and is equal to the electric moment per unit volume of the dielectric.

1.3.4 Magnetic polarization of matter (magnet)

• An external magnetic field produces a massive orienting effect on the magnetic moments of all particles and brings the substance into a state of magnetic polarization or magnetization. The degree of magnetic polarization of a substance is characterized by the magnetization vector

This value is local, since the total magnetic moment does not refer to the entire substance, but to its elementary part by volume. With uniform magnetization, the vector has the same value at all points of the substance and is equal to the magnetic moment per unit volume of the magnet.

1.4 Boundary phenomena caused by the polarization of matter

1.4.1 Is a volumetric macroscopic charge possible inside a polarized substance?

• A single electric dipole is electrically neutral and has an overall zero charge. Similarly, an imaginary magnetic dipole has an overall zero magnetic charge. Any macroscopic set of dipoles for any orientation will also have a zero charge. Therefore, both homogeneous and inhomogeneous polarization of a substance does not lead to the formation of an internal volumetric macroscopic charge either in dielectrics or in magnets.

1.4.2 Is a macroscopic molecular current possible inside a polarized magnet?

According to the Ampère model, a magnet is considered as a macroscopic set of molecular coils of current, the magnetic moments of which, during polarization, are oriented in the direction of the external magnetic field, and the planes of the coils are perpendicular to the field. In this case, the molecular currents in the contacting coils are directed oppositely throughout the entire volume of the magnet. For this reason, a nonzero net molecular current inside a magnet is impossible.

1.4.3 Localization of bound charges and bound molecular currents on the body surface

• Bound charges, both electric and magnetic, as well as molecular currents, are concentrated only at the boundary of the substance. For simplicity, it is convenient to choose a body with the simplest form of a closed boundary surface

where is the part of the surface perpendicular to the direction of polarization, and is parallel to this direction. With some approximation, this corresponds to a body in the form of a disk or a cylinder, polarized along its axis. Then is the surface of the ends of the cylinder or disk, and is their lateral surface. It is obvious that bound charges, both electric and magnetic, can concentrate only on the surface, but they will not be on the surface. On the contrary, the molecular current can concentrate only on the surface and it will not be on the surface (Fig. 1.4.3).

Rice. 1.4.3

a) polarized d and an electrician (compare accurate only on pov.)

b) polarized ma g netik (concentrated on chenny only on pov.)

1.4.4 Modeling of a polarized body by a hollow cavity.

• Since the field-forming bound charges and molecular currents are concentrated only on the body boundary, and there are none inside the body, when calculating the field, the internal space of the body within its boundaries can be considered as an empty cavity free from charges and currents. The exception applies only to the case when the body is inhomogeneous in its structure and properties, due to which

• field-forming sources may appear inside the body.

1.5 Field equation in a dielectric

1.5.1 Relation of the polarization vector to the surface density of bound charges

• At the ends of a homogeneous polarized dielectric in the form of a cylinder, bound charges are formed and transform it into a macroscopic dipole with its own electric moment, the modulus of which

where is the distance between the charges, and is the surface density of the bound charge. It is obvious that the ratio

expresses the polarization of a unit volume of the dielectric, and this, by definition, coincides with the absolute value of the polarization vector. From the above expressions it follows

This connection remains in force if the cylinder is turned into a disk. Then the end surfaces with bound charges can be considered as a flat capacitor.

Rice. 1.5.2

Polarizing field of free charges and depolarizing field of bound charges in a polarized diel to the trick

1.5.2 Field equation in a dielectric

If an unpolarized disk-shaped dielectric is introduced into a charged flat capacitor with free charges and on its plates, then it will undergo polarization with the formation of bound charges and on its surfaces. A capacitor in a capacitor or a double capacitor is formed (Fig. 1.5.2). In this case, a capacitor on free charges creates a third-party polarizing field in the dielectric, and a capacitor on bound charges creates an opposite depolarizing field, respectively.

In connection with opposite direction fields and the resulting field can be represented by their difference

Taking into account that, and considering the product as non-force, different from E q characteristic of the external field, i.e., assuming that

find the field equation in scalar form

where,

Since D , E and P are the modules of vectors parallel to each other, then the field equation in the dielectric can be represented finally in vector form

where , and are, respectively, the vector of electric induction (electric displacement), the intensity of the resulting electric field and the polarization vector of the dielectric.

1.5.3 Remark on the field equation in d and electrics

It should be emphasized that the choice of a dielectric in the form of a flat thin disk placed in a side homogeneous field of a flat capacitor provided such a physical situation when the resulting field turned out to be collinear to the side field:

, .

Under such conditions, the electric displacement vector is determined by the expression

and can be considered as a non-force characteristic of the external field. But a situation is possible when the resulting field is not collenial to the third-party field

, .

In this case, the electric displacement vector is no longer a characteristic of the external field, since

, .

Thus, one can generalize

At

At

Obviously, in the case when, the vector can be considered as a notation for the sum. In this amount, since

Therefore, in the general case, when, all three vectors and have the same direction, as in the particular case, when.

1.5.4 Dielectric constant and dielectric susceptibility of a dielectric

• The physical meaning of the permittivity and dielectric susceptibility follows from their definition

, .

Bound charges are the response of a dielectric to the action of third-party free charges. Material values

characteristics of the dielectric and are determined by this response. The closer to the more and the more.

1. Field equation in a magnet

1.6.1 Relation of the magnetization vector to molecular currents

• On the side surface of a polarized magnet in the form of a long cylinder, a common molecular current is formed, which turns it into a macroscopic magnetic dipole with its own magnetic moment, the modulus of which

where is the area of ​​the end face of the cylinder. The magnetization of a unit volume of a magnet coincides by definition with the absolute value of the magnetization vector:

where is the length of the cylinder. Thus:

1.6.2 The field equation in a magnet.

• If an unmagnetized magnet in the form of an equally long cylinder is introduced into a long conductive solenoid with a common current in all turns, then it will undergo magnetization with the excitation of a common Ampere current on its side surface. A solenoid is formed in a solenoid - Ampere in a conductive one (Fig. 1.6.2.).

Rice. 1.6.2

Magnetic polarization of a magnet. The external magnetic field sol e noida in z awakens in the magnet d about additional magnetic field of the same direction

Each of the solenoids creates its own magnetic field of the same direction, respectively, and, at the same time

, .

H value can be considered as different from e power characteristic of an external magnetic field, o b wired with a conductive solenoid. Although H ex which is not a force characteristic, it is commonly called the magnetic field strength.

Since and coincide in direction, then to The resulting magnetic field will be determined by the sum

Therefore, one can write

where,

B , H and H value are modules of vectors parallel to each other, so the field equation in a magnet can be represented finally in vector form

where, and are, respectively, the vector of magnetic induction of the resulting magnetic field, the vector of the strength of the external field, and the magnetization vector of the magnet.

1.6.3 Remarks on the field equation in a magnet

• It should be specially emphasized that the choice of a magnet in pho R me of a long rod placed in a third-party d magnetic field of a long conductive solenoid, such a physical situation was provided when the result b the tuning field turned out to be collinear to the external field:

, .

Under such conditions, the magnetic field strength vector about la can be considered as a non-force characteristic and external magnetic field and determined by the expression but by marriage

,
where is the current in a separate coil of the solenoid, N – total number in and t cov, n - their linear density.

But another situation is possible when

This is facilitated by the lack of collinearity between p e resulting and third-party fields. In all cases, to about where, the value should be understood as the designation e difference

1. Comparison of the formal and physical content of the material field equations in dielectrics and magnets

1.7.1 Force properties of electric and magnetic fields

• Electric and magnetic fields manifest themselves physically e Ski like fields of force. Each of them is capable of exerting a force action on an electric charge. t wie, respectively

where is the electric field strength, is the magnetic induction t field.

According to the indicated forces, the vectors and are easily determined in vacuum conditions. In the material environment, the vectors and also retain their power content, since the polar and zation of matter and is a consequence of the force or and orienting action of these fields on the moments of particles and, accordingly, but

, .

1.7.2 Analogues in ma erial equations

• Vectors and are the power characteristics of the electric e and magnetic fields and are analogous in their meaning. Vectors and are also analogues, defined e which determine the state of polarization, respectively, the dielectric and kov and magnets. The vectors and - are analogues in the sense that they express in a convenient form the relationship m e I wait by force fields both in the material environment and with about standing of its polarization and. In other words, and express the connection between fields and “non-fields”. Analog relations between the values ​​of constitutive equations can be visually represented by the expression and zhenami:

1.7.3 The special meaning of vectors and in terms of collinear o f the resultant and extrinsic fields

In the general case, in the absence of collinearity, when and, the vectors and are not characteristic e ristikami third-party fields, as and or otherwise and. Only in a particular case, about when the resulting and third-party fields are collinear, the condition and is satisfied, under which the vectors and become non-strong about vymi characteristics of third-party fields. In Fig.1.7.3 pr and additional explanations are given for this case.

Fig.1.7.3

Scheme of the semantic meaning of vector quantities in mater and cal equations under colloquial and non-arities

Resulting fields in matter, their sources and their power characteristics

Field sources: Field sources: currents

Free and bound conductivities and coupled

Charges together molecular currents together

Field sources: Field sources:

Free external external currents

Charges () conduction ()

Extraneous fields in matter, their sources and their

Non-force characteristics

1.7.4 Field in vacuum

In a vacuum, there is no matter, and polarization is like an electric e sky and magnetic are excluded, i.e., and, as well as and. Material equations and take their particular form for vacuum

Under vacuum conditions, the vectors and characterize the h fields, but one and the same electric field. Similarly, one and the same magnetic field is characterized by vectors and. This feature is also fulfilled in many material media, in particular, in gases, in which .

1.7.5 Electric and magnetic constants

• Electric and magnetic constants and are related to the speed of light by the relation

Their numerical values:

1.8 Electrical and magnetic characteristics of matter in material ur in opinions

1.8.1- Characteristics of the substance in the basic material equations

• Among the main ones, there are three material equations

, .

The quantities, and in these equations are character e ristics of matter, respectively, dielectrics, magnets and conductive media. Insofar as

then the characteristics of the substance should also include the dielectric and magnetic susceptibilities, respectively, and.

1.8.2 Dielectric constant

As the permittivity of a dielectric substance, its permeability is taken under conditions when the resulting and external fields in it are collinear (). In this case, it takes its maximum possible value and is expressed by the simplest expression

about how many fields and are reliably controlled by an expert and cop. Thus, it shows how many times less or, in other words, how many times the dielectric polarization weakens the external field in it.

Collinearity conditions and necessary for defining e division, requires a dielectric body certain form. In particular, it can be a thin flat disk in a side field perpendicular to the plane. from the disc bone.

1.8.3 Magnetic permeability

• As the magnetic permeability of a magnetic substance, its permeability is taken under such conditions when the resulting and external fields in it are collinear R us(). In this case, it takes its maximum b but the possible value and is determined by a simple expression and eat

In addition, in this case it is easy to find from experience, about since the induction of the field of a magnetized magnet and and n The induction of an external magnetizing field is reliably controlled by experiment. Thus, the magnet and magnetization by an external magnetic field about leads to excitation in a magnet of a stronger result and of the driving magnetic field. It shows how many times the latter is superior to the former. Deputy e Note that magnetics are characterized by the values ​​and even. The exception is diamagnets, for which about ryh.

The condition of collinearity and, necessary for the definition e fission, requires a certain shape from the body of a magnet R we. It should be a long thin rod in the side n field parallel to the axis of the rod.

1.8.4 Electrical conductivity

• The dependence of the current density in a conducting medium on n a electric field strength in it is determined by m a terial equation

The specific electrical conductivity of the medium, as its material characteristic, can be determined by the expression

in which the quantities and are controlled experimentally n volume.

1.9 Flow of a vector field through a surface. Vector field divergence

1.9.1 Vector field

• Electric and magnetic fields are vector fields and they can formally be represented by one vector field of some vector, implying that the vector is generalized: The vector field of a vector is a region of space, each point of which has its own value and its own direction of this vector. A vector field can be represented by a set of vectors at points, but the field can be more visually represented by a set of directed vector lines, each of which is constructed so that at any point the vector is tangentially directed (Fig. 1.9.1). In this case, the density of vector lines can reflect the intensity of the vector field in local regions of space. For this, it is necessary that in the local region the density of vector lines be equal to the value of the vector in this region, i.e.

where is the area transverse to the lines, and is the number of lines passing through it.

Rice. 1.9.1

Image of a vector field by a set of vectors about ditch, or a set of directed vector lines

1.9.2 Vector flow through a surface

• The elementary flow of a vector is the flow of vector lines through the area i.e.

If the area is not transverse to the lines of the vector, then

where is the unit normal to the area a.

The elementary flux of a vector should be understood as the elementary electric and magnetic fluxes of vectors, namely

The vector flow through an open surface is composed of elementary flows and is determined by the integral

The flow of a vector through a closed surface is determined by a similar integral, only over the entire closed surface

where it is taken into account that are the outer vectors on the elements of the surface.

The vector flux should be understood as the electric and magnetic fluxes of vectors, namely

As an example in Fig. 1.8.2 shows the expression for the magnetic flux.

Rice. 1.9.2

Magnetic flux through the surface: element n container, open and closed

1.9.3 Electrical flow through a closed surface

• The flow of vectors and through a closed surface of arbitrary shape is determined by the integrals

Fluxes and are scalar quantities; in the SI system, and are measured, respectively.

1.9.4 Magnetic flux through a closed surface

• Similarly, the flow of vectors and through a closed surface of arbitrary shape is determined by the integrals

Fluxes and are also scalar quantities; in the SI system, and are measured, respectively.

1.9.5 Divergence of a vector in a vector field

• Divergence is a local scalar characteristic of a vector field and determines the presence or absence of singular points in it. These are the points at which vector lines either originate or disappear, i.e. are running out. Thus, divergence defines local sources or local sinks (“absorbers”) of vector lines in a vector field. Mathematically, the divergence of a vector is defined by a simple expression:

where is a closed surface, and is a volume bounded by the same surface. Obvious conclusions follow from the divergence expression, namely, if - then vector lines are generated in the local area of ​​the field (at its separate point); if - then the lines end in the local area of ​​​​the field (at its separate point);

If - then the lines pass through the “transit” through the local area of ​​the field or through its separate point.

1.9.6 Divergence in electric and magnetic fields

• The divergence of a vector should be understood as the divergence of vectors in the electric and magnetic fields, respectively, i.e.

The concept of divergence has a mathematical content. It indicates where the source of the field is located, but does not give information about what it represents physically.

1.10 Maxwell's equations about the relationship of electric and magnetic fields with their charge sources

1.10.1 Electric field of free charges

• The electric field in a dielectric is determined by the constitutive equations

The primary source of the appearance of all quantities in this equation is a free third-party charge. As a result of its action on the dielectric, the bound charge and its field are excited, and the sum forms the electric displacement vector:

The direct relationship between and is determined by the Maxwell equation and is fundamental in its significance.

1.10.2 Maxwell's equation in integral form about the connection of a vector with a free external charge

• The electric field associated with a free charge is completely determined by its flow through a closed surface when the charge itself is inside it. For simplicity, it is appropriate to take a sphere as a closed surface, and a point charge at the center of the sphere as a free charge. Then on all elements of the sphere and, which simplifies the calculation of the flow:

For a point charge

Thus

This equation follows from Coulomb's law. It follows from the Gauss theorem that it remains valid for any form of a closed surface and for any number of free charges in it. It has also been proven that it retains its form when charges move inside the surface and even when radiation occurs through it. When the above equation is called Maxwell's equation, then all of these generalizations are meant.

1.10.3 Maxwell's equation in differential form on local charge sources of electric field

• Maxwell's differential equation follows from the integral one by limiting the volume of a closed surface and passing to the concept of divergence

This leads to the differential form of Maxwell's equation

It follows from it that the points of space at which the free charge density are singular points of the vector electric field. At these points, the vector lines originate if and disappear (end) if, and also “transit” through any point, if at it (Fig. 1.9.3).

Rice. 1.9.3

Flow and vector divergence

1.10.4 Maxwell's equation in integral form about the vector flux through a closed surface in a magnetic field

• The equation for the flow of a vector through a closed surface would be a complete analog of the equation for the flow of a vector through a closed surface if, like a free electric charge, there was a free magnetic charge. But it does not exist, it cannot be detected in a closed surface in any physical situation. So

There are no exceptions to this Maxwell equation.

1.10.5 Maxwell's equation in differential form on the absence of charge sources of a magnetic field

• The absence of free magnetic charges excludes any concept of their density, therefore Maxwell's equation in differential form states that

those. that there are no charge sources of the magnetic field. This means that there are no singular points in the magnetic field where the vector lines of the vector would begin or end. These lines are continuous throughout the space of their existence and can only be closed lines.

1.11 Vortex vector field. Circulation and rotor in a vortex field

1.11.1 Main characteristics of the vortex field

• Each of the vectors can form a vortex field with the same characteristic features, which are enough to consider using the example of one generalized vector. The field of a vector is considered vortex if all its vector lines are closed on themselves, while the closed lines do not touch or intersect. The main characteristics of the vortex field are the circulation of the vector in a closed loop and the rotor of this vector at a given point in the field.

1.11.2 Vector circulation in a closed loop in a vortex field

• The contour in the form of an arbitrary closed line covers a certain area of ​​the vector field of the vector. By the circulation of a vector along the contour, we mean the integral

where are the vector elements of the length of the contour itself, coinciding in direction with the direction of its bypass. This integral carries information about the most important thing: whether or not the vector field is vortex in the region that is limited by the contour.

So non-zero circulation means that the field is vortex within the contour and its source is inside the contour, while zero circulation indicates the absence of a vortex field source in the contour, and also that the field is potential within the contour, i.e. non-vortex. The circulation of the vector along different contours, covering the same source of the vortex field, has the same value, therefore, the circulation integral does not depend on the shape and size of the contour containing this source. If a closed vector line of the field itself is chosen as a contour, then the circulation of the vector along it is always non-zero, since it always contains within itself the source of the vortex field. It is essential that the circulation integral along the vector line of the vortex field and the integral along any contour enclosing the same source of the vortex field have the same value. On Fig. 1.11.2 provides examples of these situations.

Rice. 1.11.2

Vortex field vector

The field vector lines are concentric circles with a common center at the source of the vortex field at a point. Meaning of compasses I of the integral:

1.11.3 Vector rotor at the point of the vortex field

• The vector circulation only indicates the presence of a vortex field source within the closed loop, while the curl of the vector field determines the position of this source locally, i.e. at a specific point. The rotor, unlike circulation, is a vector quantity and is mathematically determined by a simple expression

where is the area of ​​the surface bounded by the circulation contour, which, for simplicity, is taken to be a closed vector line of the field itself, and is a unit right-handed vector normal to the plane of the vector line. On Fig. 1.11.3 provides an illustrative explanation of the definition of a rotor. Obvious conclusions follow from the rotor expression: not all points of the vortex field are its sources. So if - then in the local area (at the point) there is a source of the vortex field (there is a source of the vortex), if - then in the local area (at the point) there is no source of the vortex field (there is no source of the vortex).

Rice. 1.11.3

Vortex field rotor

1.11.4 Circulation and vector curl in vortex electric and magnetic fields

• All the justifications of the circulation and the rotor of the vector with the same mathematical formality refer to the vectors of the electric field and, as well as to the vectors of the magnetic field and:

1.12 Maxwell's equation about the connection of a vortex magnetic field with its vortex sources

1.12.1 Electrical conduction current

• If under the action of a constant electric field a constant electric current is maintained in the medium, then it is the conduction current, and the medium is conductive. Conductive media include metals, semiconductors, electrolytes, and plasma. The conductive medium is characterized by resistivity and conductivity (the reciprocal of resistivity). The current density and current are determined by the expressions

where is the cross-sectional surface of the conductor (conducting medium), is the flat cross-sectional area of ​​the conductor.

1.12.2 Eddy magnetic field of conduction current

• The conduction current is the source of the vortex magnetic field, and this should be considered as an initial physically substantiated fact. In the case of a thin rectilinear conductor with current, the vector lines of the vectors and lie in a plane transverse to the conductor and take the form of concentric circles with a right-handed orientation of the vector lines with respect to the current direction. At each point of the circular line the radius of the vector and have constant numerical values

1.12.3 Circulation of a vector in a vortex magnetic field of conduction current

• To simplify, it is convenient to choose a closed circular line of the vector itself as the circulation contour of the vector (Fig.1.12.3). Then on each element of the contour and this leads to a simple expression for the circulation integral

For linear conduction current

This implies the fundamental equation in integral form

The equation remains valid for any form of the circulation circuit, even when it covers not one, but several identically or differently directed conduction currents, i.e. when

Fig.1.12.3

Vortex magnetic field of linear conduction current

1.12.4 Vector rotor in a vortex magnetic field of conduction current

• From Section 1.11.3 it follows that the rotor of the vector is obtained from its circulation by limiting the area limited by the contour, taking into account the fact that in this case there is a simultaneous transition from current to its density

Thus

It follows from this fundamental equation in differential form that only that local region of space where there is a conduction current density can be considered as a source of a vortex magnetic field. In this case, and In the same regions of the magnetic field, including the vortex one, where also and, i.e. in such regions there can be no source of the vortex field.

Thus, the conduction current is the source of the vortex magnetic field, and the current density is

His is a local source. But the same source, in addition to the conduction current, is the displacement current, the essence of which will be clarified below.

1.12.5 The principle of closed electric current

• Capacitor included in the circuit alternating current, breaks its conductive part, but does not break the alternating current in it. The electric current remains closed. Conduction current flowing through the conductor

part of the circuit, finds its continuation in a different form, namely in the form of a displacement current inside the capacitor, where there is no conductive medium, and there can be no conduction current (Fig. 1.12.5). Thus, in magnitude and direction, the displacement current and the conduction current must match, and are determined by the change in the free charge on the capacitor plates

Rice. 1.12.5

Displacement current on not pr about water section of the circuit (in the capacitor)

1.12.6 Bias current

• For the first time, Maxwell pointed out the existence of a displacement current, based on the principle of current continuity in all sections of a closed current circuit. Considering that for a flat capacitor,

and also that

the bias current can be represented by the expression

Thus, the displacement current is not associated with the directed movement of free charges inside the capacitor, where they

no, but with a change in the displacement flux inside the capacitor. You can also express the displacement current density

As can be seen, the direction of the displacement current density is determined not by the direction of the vector, but by the change in this vector. This is very significant, because and in a condenser they have one direction only when the modulus increases, whereas when the modulus decreases, the vector is opposite, although the latter retains its former direction. It is the vector that gives the displacement current a direction that is consistent with the direction of the conduction current in the conductor part of the circuit.

1.12.7 Displacement current components

• Based on the field equation in a dielectric (Section 1.5.2), the physics of the displacement current can be further explored. From transformation

it can be seen that the bias current density consists of two components

One of the components is in no way connected with the movement of charges and is generated only by a change in the electric field in the dielectric. The other component is generated by a change in the dielectric polarization vector and is associated with the movement of charges inside the dielectric, but not free, but bound in dipole structures. The alternating field excites the reorientation of the dipoles and the displacement of their poles, i.e. associated charges. Essentially, this process of mass displacement of bound charges excites a special polarization current in the dielectric.

1.12.8 Eddy magnetic field of displacement current

• Although in its physical nature the displacement current is significantly different from the conduction current, it, like the conduction current, excites a vortex magnetic field and is its source. At present, this conclusion is accepted as the initial experimentally substantiated fact.

Then, by analogy with the conduction current, we can write the same fundamental equations for the displacement current

1.12.9 Full current eddy magnetic field

• If in the medium, along with the excitation of the displacement current, the conduction current is also excited, then the magnetic field will be determined by the total current and the total current density, respectively

The total magnetic field of the total current is also eddy, and the current itself is its source.

1.12.10 Maxwell's equations about the vortex magnetic field of the total current

• By analogy with the equations for the eddy magnetic fields of the conduction current and displacement current, similar fundamental equations remain valid for the eddy magnetic field of the total current

The first of these equations is called the Maxwell equation in integral form and, taking into account (1.12.6) and (1.12.9), is written as

The second of the equations is called the Maxwell equation in differential form and, taking into account (1.12.6) and (1.12.9), is written as

1.12.11 Alternating electric field as a source of a vortex magnetic field in vacuum

• If a dielectric is removed from a capacitor connected to an alternating current circuit and a vacuum is created between its plates, then in this case there is no break in the current in the circuit. This means that in the empty space between the capacitor plates there is a displacement current as a continuation of the conduction current in the conductor part of a closed current circuit. In empty space, the conduction current and the polarization of matter are excluded. Assuming and, and taking into account that for vacuum, the equations

Maxwell will take the form

Thus, from the consistent development of Maxwell's concept of the closedness of the current circuit and the existence of a displacement current, the most important fundamental physical conclusion follows: an alternating electric field excites a vortex magnetic field. On Fig. 1.12.11 illustrates this conclusion on the example of a uniform alternating electric field.

Rice. 1.12.11

Excitation of a vortex magnetic field per e alternating electric field (bias current)

1.13 Maxwell's equation about the connection of the vortex electric field with its vortex sources

1.13.1 Faraday's law of electromagnetic induction

• AT closed loop conductor under the action of a variable magnetic flux, an induction emf is excited, proportional to the rate of its change. The law was established by Faraday in 1831. At that time, it was believed that this law manifests itself only in a material circuit, when the circuit is a conductor. In this case, the induction emf can be considered as the sum of voltage drops dU on all contour elements, i.e.

Therefore, Faraday's law for a material conductor circuit can be represented as

1.13.2 Maxwell's law of electromagnetic induction

• Under the action of the induction EMF in a closed conductor circuit, an inductive conduction current arises, which is possible only under the action of an electric field. Then, in terms of the intensity of this field, one can also express the voltage drop dU on the elements of the circuit and in general EMF in the circuit

after which Faraday's law can be represented as

where the direction of the circuit element corresponds to the direction of the induction current in the circuit.

1.13.3 Excitation of a vortex electric field by an alternating magnetic flux in a conductor circuit

• From the Maxwellian interpretation of the law of electromagnetic induction, it follows that an alternating magnetic flux excites an electric field in a conductor circuit and that the circulation of the strength of this field along the circuit is different from zero

But this, as follows from (1.11.2.), is the main sign that the vector field in the conductor circuit is vortex and that the source of this vortex field is an alternating magnetic flux.

1. Maxwell's equations for a vortex electric field

Further development of the Maxwellian interpretation of Faraday's law is associated with the assumption that under the action of an alternating magnetic flux, a vortex electric field is excited not only in the conductor circuit, but also outside it in the surrounding space. Circuitis simply present in the vortex electric field, and it is this that creates the induction EMF in the circuit. A variable magnetic flux in the absence of a conductor circuit excites a vortex electric field as well as in the presence of a circuit. Thus, the variable magnetic flux is the source of the vortex electric field, and this should be considered as the initial foundation

tal fact that found a physical experimental substantiation. In this case, in accordance with (1.13.2.), the fundamental relationship between the alternating magnetic and vortex electric fields is reduced to Maxwell's equations

Figure 1.13.4. an illustration of the excitation of a vortex electric field by a variable uniform magnetic flux is given.

Rice. 1.13.4

Excitation of the vortex electric field p e belt magnetic field

1. Complete system of Maxwell equations

1.14.1 Maxwell's equations and their form

• Maxwell's equations express the relationship of electric, magnetic and electromagnetic fields with their sources. and kami - with an arbitrary system of charges and currents. With a comprehensive physical content, four ma to Swell's equations turned out to be sufficient to create a of the fundamental scientific foundations of the classical element to trodynamics, as well as the foundations of the electromagnetic theory of St. e that. Electromagnetic fields are vector fields, due to which Maxwell's equations are expressed "in the language" of vecto R analysis. In the differential form of notation, they are local in nature, since they establish a connection between the fields and their sources at a separate arbitrary point in the environment. In the integral form of notation, they are defined e make communication not in a separate point of the environment, but in the whole area

environment, limited or closed surface S , or a closed contour line L . Maxwell's equations with the same base are applicable to both homogeneous and inhomogeneous fields, while taking into account that in n about In the latter case, the time derivatives of the vectors in e masks become private production d nym.

1. Maxwell's first equation

Maxwell's first equations are the fundamental physical e sky law, according to which the source of the vortex ma G filament field can only be currents, including current pr about conductance, displacement current and apparent current. The connection of the vortex magnetic field with its sources is expressed in two ways - by an equation in integral form or cheers in in the differential form, respectively n but:

Circulation of a vector along an arbitrary closed loop n tour line L equal to full current I + I cm passing through the surface, and defined by the contour L .

Every point in the environment is a local source of vi X magnetic field, if only the density in it is l current.

In the absence of conduction current, when I = 0 and the equations are simplified accordingly:

It follows from them that the source of the vortex magnetic field is the displacement current or, in fact, an alternating electric field.

1. Maxwell's second equation

Maxwell's second equation is a fundamental physical law, according to which only an alternating magnetic field can be a source of a vortex electric field. The relationship between the vortex electric field and the alternating magnetic field is expressed by an integral or differential equation, respectively:

Vector circulation along an arbitrary closed contour line L is equal to the rate of change of the magnetic flux through the surface bounded by the contour, taken with the opposite sign L .Each point of the medium is a local source of a vortex electric field if the vector at this point is variable

Maxwell's second equation is formally not similar to the first one, and this is due to the absence of free magnetic charges and magnetic currents in nature. With their hypothetical presence, similarity would take place, and the equations would look like:

, .

In this hypothetical case, the magnetic current I m would be the source of the vortex electric field. But p about Since there is no magnetic current, the only real source of the eddy electric field can only be an alternating magnetic field. However, Maxwell's second equation can be similar to the first at an hour t otherwise, when the former refers to the displacement current in the absence of conduction current, i.e., when the eddy current G the filament field is excited only by an alternating electric e sky field. Then

1.14.4 Third level

Maxwell's third equation is the fundamental physical and the logical law of the relationship between the electric field and its charge about source. The law defines the connection of the electric field in the medium with external free electric charges and expresses this connection mathematically in the integral b noah or differential forms, respectively n but:

Vector flow through an arbitrary closed surface S equal to free charge q inside this surface, while the charge can be constant or variable, at rest or move, be point or distributed. A single point in the environment whereρ ≠0, is a local source (or sink) of the vector field.

1. Maxwell's fourth equation

• Maxwell's fourth equation is a fundamental physical law, according to which the magnetic field does not have its charge source in the form of a magnetic charge due to its real absence in nature. Mathematically, this fact is expressed by the integral or differential equations, respectively

Vector flow through an arbitrary closed surface xness S is always zero. This means that, passing through a closed X ness, the magnetic flux inside it does not undergo any changes for any physical and cal situations, i.e. magnetic p about current passes through the circuit at

Thuyu surface "transit". This applies not only to the flow, but also to a separate vector line of force, since local magnetic charges do not exist anywhere. For this reason, the line of force of the vector cannot be interrupted anywhere, it is continuous everywhere, which means that to chickpeas on itself. From the fourth Maxwell equation e blows the conclusion that the magnetic field cannot be about potential, it can only be a whirlwind e vym.

1.14.6 Virtual magnetic charges and magnetic currents in Maxwell's symmetric equations

• Maxwell's equations are not symmetrical both in terms of charge sources of the field and in terms of sources of the vortex field, which is directly related to the absence of magnetic charges and magnetic currents, which in reality do not exist. In this respect, Maxwell's equations are realistic. However, Maxwell's asymmetric equations take on a symmetrical form by formally introducing into them a magnetic charge and a magnetic current with a density, respectively, and. Then the system of equations takes the form

where the sign ''– , reflects only that the direction of the vortex magnetic field corresponds to the right screw, and the electric one to the left. Despite the artificial achievement of symmetry, these equations, nevertheless, turned out to be useful for substantiating calculation models, for example, for calculating the radiation of electromagnetic waves from radiating devices - antennas. So, instead of considering the real radiating source itself, an abstract radiating surface with magnetic currents enclosing it is considered. At the same time, in the final results on the calculation of radiation magnetic currents are excluded, they appear only in intermediate calculations as virtual currents. One can refer to an analogue of this method in optics, namely the Huygens-Fresnel method, in which the real source of a light wave is also replaced by a radiating surface, on which point sources of secondary waves are concentrated.

1.14.7 Significance of Maxwell's Equations

• Maxwell's equations constitute the fundamental scientific basis all electrodynamics. Based on them, the existence of electromagnetic waves was proved and the electromagnetic nature of light was substantiated. On the basis of Maxwell's equations, the scientific unity of electricity and magnetism, electrodynamics and wave optics was achieved.

It is appropriate to quote the words of the famous German physicist G. Hertz about Maxwell's equations:

It is impossible to study this amazing theory without experiencing at times such a feeling that mathematical formulas have a life of their own, have their own mind - it seems that these formulas are smarter than us, smarter even than the author himself, as if they give us more than they once contained “.

1.14.8 Solution of Maxwell's equations

• Maxwell's equations are compiled for a specific electromagnetic problem, in which, on the basis of a physical analysis of the situation, the initial features of the fields and their sources are identified in advance and, at the same time, material equations are established for baseline tasks. The mathematical solution of the problem is achieved only on the basis of a joint system of Maxwell's equations and material equations.

1.15 Stationary electromagnetic processes

1.15.1 Stationarity condition

• Stationary electromagnetic processes are realized at time-invariant magnetic and electric fields and constant currents, for which it is necessary that there are no time derivatives in Maxwell's equations, i.e.:

1.15.2 Maxwell's equations for stationary processes

• Maxwell's equations, after excluding time derivatives from them, take the form of stationary equations

In essence, these are the basic laws of a vast class of stationary electromagnetic processes. One part of this class belongs to electrostatics, another to magnetostatics, and a third to current statics (direct current).

1.15.3 Electrostatics

Electrostatics studies a constant electric field in a vacuum, in dielectrics and in conductors in the absence of a magnetic field and an electric current. If we exclude the magnetic field and current from the stationary equations, then the Maxwell equations for electrostatics take the form

1. magneto-statics

• Magnetostatics studies a constant magnetic field in vacuum and magnets, as well as a direct current magnetic field. Magnetostatic phenomena are considered in the absence of an electric field and in the absence of free macroscopic electric charges. If they are excluded from the stationary equations, then Maxwell's equations for mannitostatics take the form:

1.15.5 Current statistics (DC)

• Current statics includes electromagnetic processes in circuits made of conductive materials, in which a constant electric current is excited under the action of macroscopic electric charges and electric fields, while the magnetic field of the current, as related to magnetostatics, is not considered. In this case, for current statics, two stationary Maxwell equations related to electric charges and electric fields are sufficient:

1.16 Non-stationary electromagnetic processes

1.16.1 Non-stationarity condition

• The non-stationarity of electromagnetic processes, both in vacuum and in matter, is due to the variability in time of electric and magnetic fields. Variables

fields excite alternating conduction currents and alternating displacement currents. Thus, for non-stationary processes

those. all quantities are variables.

1.16.2 Maxwell's equations for non-stationary processes

• The whole vast variety of non-stationary electromagnetic processes obeys the non-stationary Maxwell equations in their full form

where is the sum

means full current, i.e. conduction current and displacement current.

1.16.3 Main groups of non-stationary processes

• Non-stationary electromagnetic processes are divided into significantly different groups depending on the ratio between the conduction current and the displacement current, or rather between their amplitude values ​​and, since the currents themselves are variable and usually change according to a harmonic law with a cyclic frequency. Therefore, the ratio between the amplitudes and will essentially depend on the frequency and on the properties of the substance in which the electromagnetic process is excited.

Possible options:

1.16.4 Non-stationary processes in conducting media (in metals)

• An alternating electric field in a conducting medium, especially in a metal, excites an alternating conduction current so much greater than the displacement current that the latter can be neglected even at very high frequencies, which means

The non-stationary Maxwell equations for a conducting medium (for metals) take the form

where all quantities are variable. It is essential that the alternating current magnetic field remains eddy and is related to the current in the same way as in the stationary mode.

1.16.5 Non-stationary processes in non-conducting dielectrics

• Conduction current in non-conductive dielectrics is excluded, only displacement current remains possible, so that

Thus, the non-stationary Maxwell equations for non-conducting dielectrics take the form

1.16.6 Non-stationary processes in vacuum

• In vacuum, both free macroscopic electric charges and conduction currents are excluded, but the displacement current remains possible, so that

wherein

Thus, the non-stationary Maxwell equations for vortex fields in vacuum take the form

They determine the formation of an electromagnetic field in the form of electromagnetic waves propagating at the speed of light. It also follows from the equations that the electromagnetic field generates itself and can exist without charges and currents.

Definition 1

Electrodynamics is a theory that considers electromagnetic processes in vacuum and various media.

Electrodynamics covers a set of processes and phenomena in which actions between charged particles play a key role, which are carried out by means of an electromagnetic field.

History of the development of electrodynamics

The history of the development of electrodynamics is the history of the evolution of traditional physical concepts. Even before the middle of the 18th century, important experimental results were established, which are due to electricity:

• repulsion and attraction;
• division of matter into insulators and conductors;
• the existence of two types of electricity.

Considerable results have also been achieved in the study of magnetism. The use of electricity began in the second half of the 18th century. The emergence of the hypothesis of electricity as a special material substance is associated with the name of Franklin (1706-1790). And in 1785, Coulomb established the law of interaction of point charges.

Volt (1745-1827) invented many electrical measuring instruments. In 1820, a law was established that determined mechanical force, with which the magnetic field acts on the electric current element. This phenomenon is called Ampère's law. Ampere also established the law of the force of several currents. In 1820 Oersted discovered magnetic action electric current. Ohm's law was established in 1826.

In physics, the hypothesis of molecular currents, which was proposed by Ampere back in 1820, is of particular importance. Faraday discovered the law of electromagnetic induction in 1831. James Clerk Maxwell (1831-1879) in 1873 set out the equations that later became the theoretical basis of electrodynamics. A consequence of Maxwell's equations is the prediction of the electromagnetic nature of light. He also predicted the possibility of the existence of electromagnetic waves.

Over time in physical science there was an idea of ​​the electromagnetic field as an independent material entity, which is a kind of carrier of electromagnetic interactions in space. Various magnetic and electrical phenomena have always aroused people's interest.

Often, the term "electrodynamics" is understood as traditional electrodynamics, which describes only the continuous properties of the electromagnetic field.

The electromagnetic field is main subject the study of electrodynamics, as well as a special kind of matter, which manifests itself when interacting with charged particles.

Popov A.S. In 1895 he invented the radio. It was it that had a key impact on the further development of technology and science. Maxwell's equations can be used to describe all electromagnetic phenomena. The equations establish the relationship of quantities that characterize magnetic and electric fields, distributing currents and charges in space.

Figure 1. Development of the doctrine of electricity. Author24 - online exchange of student papers

Formation and development of traditional electrodynamics

The key and most significant step in the development of electrodynamics was the discovery of Faraday - the phenomenon of electromagnetic induction (excitation of an electromotive force in conductors using an alternating electromagnetic field). This is what became the basis of electrical engineering.

Michael Faraday is an English physicist who was born to a blacksmith's family in London. He graduated from elementary school and worked as a paperboy from the age of 12. In 1804, he became a student of the French émigré Ribot, who encouraged Faraday's desire for self-education. In lectures, he sought to replenish his knowledge of natural sciences chemistry and physics. In 1813 he was presented with a ticket to Humphry Davy's lectures, which played a decisive role in his fate. With his help, Faraday got a position as an assistant at the Royal Institution.

Faraday's scientific activity took place at the Royal Institute, where he first helped Davy in his chemical experiments, after which he began to conduct them on his own. Faraday obtained benzene by reducing chlorine and other gases. In 1821, he discovered how a magnet rotates around a conductor with current, thus creating the first model of an electric motor.

Over the next 10 years, Faraday has been studying the relationship between magnetic and electrical phenomena. All his research was crowned with the discovery of the phenomenon of electromagnetic induction, which happened in 1831. He studied this phenomenon in detail, and also formed its basic law, during which he revealed the dependence of the induction current. Faraday also studied the phenomena of closing, opening and self-induction.

The discovery of electromagnetic induction produced scientific significance. This phenomenon underlies all alternating and direct current generators. Since Faraday constantly sought to reveal the nature of electric current, this led him to conduct experiments on the passage of current through solutions of salts, acids and alkalis. As a result of these studies, the law of electrolysis appeared, which was discovered in 1833. This year he opens a voltmeter. In 1845, Faraday discovered the phenomenon of polarization of light in a magnetic field. In this year he also discovered diamagnetism, and in 1847 paramagnetism.

Remark 1

Faraday's ideas about magnetic and electric fields had a key influence on the development of all physics. In 1832, he suggested that the propagation of electromagnetic phenomena is a wave process that occurs at a finite speed. In 1845, Faraday first used the term "electromagnetic field".

Faraday's discoveries gained wide popularity around the world. scientific world. In his honor, the British Chemical Society established the Faraday Medal, which became an honorary scientific award.

Explaining the phenomena of electromagnetic induction and having encountered difficulties, Faraday suggested the implementation of electromagnetic interactions with the help of electric and magnetic fields. All this laid the foundation for the creation of the concept of the electromagnetic field, which was framed by James Maxwell.

Maxwell's contribution to the development of electrodynamics

James Clerk Maxwell is an English physicist who was born in Edinburgh. It was under his leadership that the Cavendish Laboratory in Cambridge was created, which he headed all his life.

Maxwell's works are devoted to electrodynamics, general statistics, molecular physics, mechanics, optics, and also the theory of elasticity. He made the most significant contribution to electrodynamics and molecular physics. One of the founders of the kinetic theory of gases is Maxwell. He established the distribution functions of molecules in terms of velocities, which are based on the consideration of reverse and direct collisions; Maxwell developed the theory of transport in general view and applied it to the processes of diffusion, internal friction, heat conduction, and also introduced the concept of relaxation.

In 1867 he first showed the statistical nature of thermodynamics, and in 1878 introduced the concept of "statistical mechanics". Maxwell's most significant scientific achievement is his theory of the electromagnetic field. In his theory, he uses the new concept of " displacement current"And gives a definition of the electromagnetic field.

Remark 2

Maxwell predicts a new important effect: the existence electromagnetic radiation and electromagnetic waves in free space, as well as their propagation at the speed of light. He also formulated a theorem in the theory of elasticity, establishing the relationship between key thermophysical parameters. Maxwell develops the theory of color vision, explores the stability of Saturn's rings. He shows that the rings are not liquid or solid, they are a swarm of meteorites.

Maxwell was a famous popularizer of physical knowledge. The contents of his four electromagnetic field equations are as follows:

1. The magnetic field is generated by moving charges and an alternating electric field.
2. An electric field with closed lines of force is generated by an alternating magnetic field.
3. Magnetic field lines are always closed. This field does not have magnetic charges, which are similar to electric ones.
4. The electric field, which has open lines of force, is generated by electric charges, which are the sources of this field.

Lecture notes

Approved by the Editorial and Publishing Council of the University as lecture notes

Reviewers:

Doctor of Physical and Mathematical Sciences, Head. Department of T and EF KSTU, Professor A.A. Rodionov

Candidate of Physical and Mathematical Sciences, Head. department
General Physics KSU Yu.A. Neruchev

Candidate technical sciences, head Department of Physics, KSHA
DI. Yakirevich

Polunin V.M., Sychev G.T.

Physics. Electrostatics. Constant electric current: Lecture notes / Kursk. state tech. un-t. Kursk, 2003. 196 p.

The lecture notes are compiled in accordance with the requirements of the State Educational Standard-2000, Sample program disciplines "Physics" (2000) and work program in physics for students of engineering and technical specialties of KSTU (2000).

The presentation of the material in this paper provides for students' knowledge of physics and mathematics in the volume school curriculum, a lot of attention is paid to difficult-to-understand questions, which makes it easier for students to prepare for the exam.

The abstract of lectures on electrostatics and direct electric current is intended for students of engineering and technical specialties of all forms of education.

Il. 96. Bibliography: 11 titles.

Ó Kursk State
Technical University, 2003

Ó Polunin V.M., Sychev G.T., 2003

Introduction.. 7

Lecture 1. Electrostatics in vacuum and matter. Electric field 12

1.1. The subject of classical electrodynamics.. 12

1.2. Electric charge and its discreteness. The theory of close action. thirteen

1.3. Coulomb's law. Electric field strength. The principle of superposition of electric fields.. 16

1.4. Electrostatic field strength vector flow. 22

1.5. The Ostrogradsky-Gauss theorem for an electric field in vacuum. 24

1.6. The work of an electric field on the movement of an electric charge. Circulation of the electric field strength vector. 25

1.7. The energy of an electric charge in an electric field. 26

1.8. Potential and potential difference of the electric field. Connection of electric field strength with its potential.. 28

1.9. Equipotential surfaces.. 30

1.10. Basic equations of electrostatics in vacuum. 32

1.11. Some examples of electric fields generated by the simplest systems of electric charges. 33

Lecture 2. Conductors in an electric field .. 42

2.1. Conductors and their classification. 42

2.2. Electrostatic field in the cavity of an ideal conductor and near its surface. Electrostatic protection. Distribution of charges in the volume of the conductor and over its surface.. 43

2.3. Electric capacitance of a solitary conductor and its physical meaning. 46

2.4. Capacitors and their capacitance. 47

2.5. Capacitor connections. 51

2.6. Classification of capacitors. 54

Lecture 3. Static electric field in matter.. 55

3.1. Dielectrics. Polar and non-polar molecules. Dipole in homogeneous and inhomogeneous electric fields. 55

3.2. Free and bound (polarization) charges in dielectrics. Polarization of dielectrics. Polarization vector (polarization) 58

3.3. Field in dielectrics. electrical displacement. Dielectric susceptibility of matter. Relative permittivity of the medium. The Ostrogradsky-Gauss theorem for the flow of the electric field induction vector. 61

3.4. Conditions at the interface between two dielectrics. 63

3.5. Electrostriction. Piezoelectric effect. Ferroelectrics, their properties and applications. electrocaloric effect. 65

3.6. Basic equations of electrostatics of dielectrics. 72

Lecture 4. Electric field energy.. 75

4.1. Energy of interaction of electric charges. 75

4.2. The energy of charged conductors, a dipole in an external electric field, a dielectric body in an external electric field, a charged capacitor. 77

4.3. Electric field energy. Volumetric energy density of the electric field 81

4.4. Forces acting on macroscopic charged bodies placed in an electric field. 82

Lecture 5. Direct electric current .. 84

5.1. Constant electric current. The main actions and conditions for the existence of direct current. 84

5.2. The main characteristics of direct electric current: the value /strength/ current, current density. Third party forces.. 85

5.3. Electromotive force (EMF), voltage and potential difference. their physical meaning. Relationship between EMF, voltage and potential difference. 90

Lecture 6. Classical electronic theory of conductivity of metals. Direct current laws.. 92

6.1. Classical electronic theory of electrical conductivity of metals and its experimental justifications. Ohm's law in differential
and integrated forms. 92

6.2. Electrical resistance of conductors. Change of resistance of conductors from temperature and pressure. Superconductivity. 98

6.3. Resistance connections: series, parallel, mixed. Shunting of electrical measuring instruments. Additional resistances to electrical measuring instruments.. 104

6.4. Rules (laws) of Kirchhoff and their application to the calculation of the simplest electrical circuits 108

6.5. Joule-Lenz law in differential and integral forms. 110

6.6. Energy released in a DC circuit. Coefficient of performance (COP) of a direct current source. 112

Lecture 7. Electric current in vacuum, gases and liquids .. 115

7.1. Electric current in vacuum. Thermionic emission. 115

7.2. Secondary and field emission. 122

7.3. Electric current in gas. Ionization and recombination processes.. 124

7.4. The concept of plasma. Plasma frequency. Debye length. Plasma electrical conductivity 142

7.5. electrolytes. Electrolysis. Laws of electrolysis. 149

7.6. Electrochemical potentials.. 151

7.7. Electric current through electrolytes. Ohm's law for electrolytes. 152

Lecture 8. Electrons in crystals.. 161

8.1. Quantum theory of electrical conductivity of metals. Fermi level. Elements of the band theory of crystals. 161

8.2. The phenomenon of superconductivity from the point of view of the Fermi-Dirac theory. 170

8.3. Electrical conductivity of semiconductors. The concept of hole conductivity. Intrinsic and extrinsic semiconductors. The concept of p-n - transition. 171

8.4. Electromagnetic phenomena at the interface between media. 178

conclusion.. 193

REFERENCES.. 195

This manual is compiled on the basis of materials developed by the authors in the process of giving lectures on general physics to students of engineering and technical specialties, with a relatively small volume of classroom studies, over a long period of time.

The presence of this lecture notes for students of engineering and technical specialties will allow them and the lecturer to use lecture time more efficiently, pay more attention to difficult-to-understand questions, and make it easier for students to prepare for the exam.

Particularly in need of such a manual, in our opinion, are students of correspondence, accelerated and distance forms of education, who, starting to study physics, have insufficient skills in adequate perception of physical concepts, definitions and laws.

The presentation of the material in this work provides for students' knowledge of physics and mathematics in the scope of the school curriculum, therefore, many concepts are not disclosed in detail in it, but are used as well-known ones. In addition, this work assumes that students have already studied or are studying the corresponding mathematical apparatus (differential and integral calculus, analysis of functions, differential equations, vector algebra, series) in parallel with the course.

A feature of the manual is that the material is presented in it in a certain, non-traditional sequence, contains the necessary drawings and explanations.

Despite the small volume, the proposed manual contains a statement of issues, the knowledge of which is necessary for the study of disciplines, the foundation of which are the laws and basic provisions of physics.

The reduction in volume was achieved mainly by refusing to consider certain non-principled issues, as well as by submitting some issues for their study in the process of practical and laboratory classes.

Such issues as the band theory of metals and semiconductors, current in vacuum, gases and electrolytes are presented in sufficient detail.

The presentation of the material, with rare exceptions due to methodological considerations, is based on an experiment. The fundamental experiments that formed the basis of the modern theory of electromagnetism are described in sufficient detail.

In addition, a certain attention is paid to the explanation of the principles of measuring the basic electrical quantities, which, if possible, follows immediately after the introduction of the relevant physical concepts. However, the description of various experiments does not claim to be complete and, moreover, only concerns the principles of these experiments, since students listen to a lecture course with demonstrations and work in physical laboratories. For the same reason, most of the drawings are made in the form simple circuits and reflects only qualitative dependencies for this case without indicating units of measurement and numerical values considered quantities, which contributes to a better perception of the studied material by students.

Since at present there are problem books corresponding to the university course of physics, the inclusion specific tasks and exercises for the studied section are not provided. Therefore, only a relatively few examples are given in the lecture notes to illustrate the application of the most important laws.

The presentation is carried out in the International System of Units (SI). Unit designations physical quantities are given in terms of the basic and derived units of the system, in accordance with their definitions in the SI system.

The manual can be used by graduate students and teachers who have insufficient experience in the university.

The authors will be grateful to all who carefully review this manual and make certain comments on the merits. In addition, they will try to take into account all rational comments from fellow physicists, graduate students, and students and make appropriate corrections and additions.

Introduction

This lecture notes are devoted to one of the sections of the general course of physics, the section "Electricity", which is read to students of those specialties and forms of education in whose curricula this course is provided.

It focuses on the fact that electrical energy plays a big role in technology for the following reasons:

1. The extreme ease with which electricity is converted into other types of energy: mechanical, thermal, light and chemical.

2. Ability to transmit electricity over long distances.

3. High efficiency of electric machines and electric devices.

4. Extremely high sensitivity of electrical measuring and recording instruments and development electrical methods measurements of various non-electric quantities.

5. Exceptional features provided electrical appliances and devices for automation, remote control and production control.

6. Development of electrical, electrothermal, electrochemical, electromechanical and electromagnetic methods of material processing.

The doctrine of electricity has its own history, organically linked with the history of the development of the productive forces of society and other areas of natural science. In the history of the doctrine of electricity, three stages can be distinguished:

1. The period of accumulation of experimental facts and the establishment of basic concepts and laws.

2. The period of formation of the doctrine of the electromagnetic field.

3. The period of formation of the atomistic theory of electricity.

The origins of ideas about electricity go back to Ancient Greece. The attraction of light bodies by rubbed amber and other objects has been known to people for a long time. However, the electrical forces were completely unclear, the possibility of their practical application was not felt, so there was no incentive for systematic research in this area.

Only the discoveries of the first half of the XYIII century. force a sharp change in attitude towards electrical phenomena. Undoubtedly, this was facilitated by the invention electrical machine(second half of the XYII century), on the basis of which the possibilities of experimentation were significantly expanded.

By the middle of the XIII century. interest in electricity is growing, natural scientists from many countries are included in the research. Observation of strong electric discharges could not but lead to an analogy between an electric spark and lightning. The electrical nature of lightning was proved by the direct experiments of W. Franklin, M.V. Lomonosov, G.V. Richman (1752 - 1753). The invention of the lightning rod was the first practical application of the doctrine of electricity. This contributed to the development of a general interest in electricity, attracting new researchers to this area.

The English naturalist R. Simmer (1759) put forward a fruitful hypothesis about the nature of electricity. Developing the ideas of Dufay, Simmer concluded that bodies in ordinary states contain two kinds of electricity in equal amounts, neutralizing each other's action. Electrification causes an excess in the body of one electricity over another. An excellent confirmation of this hypothesis was the discovery of electrostatic induction by the Russian academician F. Epinus (1759).

The law of conservation of energy and matter established by Lomonosov was the greatest achievement in physics of the 18th century. The content of the conservation law discovered by Lomonosov was gradually revealed and played a great role in the development of the theory of electricity. Thus, the later discovered law of conservation of electric charges is a particular manifestation of the universal law of conservation of matter and motion.

Until the middle of the XIII century. electrical experiments continued to be purely qualitative. The first step towards a quantitative experiment was taken by Richmann, who proposed the first instrument for measurements, called the electrometer (1745). The most important stage in the development of experimental technology was the invention in 1784 by S. Coulomb of a very sensitive torsion balance, which played important role in the study of forces of various nature. This device allowed Coulomb to establish the law of interaction between magnets and electric charges (1785). Coulomb's laws served as the basis for the development of the mathematical theory of electrostatics and magnetostatics.

Further, thanks to the experiments of L. Galvani (1789) and A. Volta (1792), contact electrical phenomena were discovered, which, in turn, led to the invention of galvanic cells and to the discovery of electric current (1800).

English researchers A. Carlyle and W. Nicholson discovered that the galvanic current, passing through water, decomposes it into hydrogen and oxygen. A mutually enriching relationship has been established between physics and chemistry. Electricity acquires tremendous practical importance, which stimulates the further development of this branch of science.

Improving the design of the voltaic column leads to the discovery of new actions of electric current. In 1802 V.V. Petrov, with the help of a powerful voltaic column, receives electric arc. The Petrov arc gave rise to a number of new applications of the thermal effects of current.

With the discovery of the action of current on a magnetic needle, H. Oersted (1820) laid the foundation for a new chapter in the theory of electricity - the doctrine of the magnetic properties of current, which made it possible to include magnetism in a unified theory of electromagnetic phenomena.

The study of electric current continued to progress at an increasing pace. It was found that the magnetic effect of the current is enhanced if the conductor is coiled. This opened up the possibility of designing electromagnetic current meters.

In 1820, A. Ampère established a law by which the force of interaction of two elementary currents was determined. Based on this experimental fact, A. Ampère makes an assumption about the electrical nature of magnetism. He suggests that "electric currents ... exist around particles in iron, nickel and cobalt already before magnetization. Being, however, directed in all possible directions, they cannot cause any resulting external action, since some of them tend to attract what others push away…". This is how the hypothesis of molecular currents appeared in physics, the depth of which was revealed only in the 20th century.

In further research on electricity, the law established in 1827 by the German physicist G. Ohm and called Ohm's law became an effective tool.

During this period began scientific activity M. Faraday. Two of Faraday's discoveries are of particular importance in the history of physics: the phenomenon of electromagnetic induction (1831) and the laws of electrolysis (1834). Faraday, with these discoveries, provided the theoretical basis for many technical applications of electricity. E.Kh. Lenz on electromagnetic induction (Lenz's rule) and the establishment of a law for thermal action current (Joule-Lenz law) contributed to further practical application electricity.

It was experimentally established that electric forces act through a medium that fills the space between interacting bodies. Exploring the interaction of charged bodies, Faraday introduced the concept of electric lines of force and gave the idea of ​​magnetic and electric fields - spaces where the action of electric forces is detected. Faraday believed that electric and magnetic fields represent deformed states of some all-penetrating weightless medium - the ether.

According to Faraday, it is not the electric charge that acts on the surrounding bodies, but the lines of force associated with the charge. By this Faraday put forward the idea of ​​the theory of short-range action, according to which the action of some bodies on others is transmitted through environment at a certain speed.

In the 60s of the 19th century, D. Maxwell generalized Faraday's theory of electrical and magnetic fields and created a unified theory of the electromagnetic field. The main content of this theory lies in Maxwell's equations, which play the same role in electromagnetism as Newton's laws in mechanics.

It should be noted the great importance of the work of a number of Russian physicists of the late 19th century. on experimental confirmation of Maxwell's theory. Among such studies, the experiments of P.N. Lebedev on the detection and measurement of light pressure (1901).

Until the end of the 19th century. electricity was represented as a weightless liquid. The question of whether electricity is discrete or continuous required the analysis of experimental material and the setting up of new experiments. The idea of ​​the discreteness of electricity can be seen in discovered by Faraday the laws of electrolysis. Based on these laws, the German physicist G. Helmholtz (1881) suggested the existence of the smallest portions of the electric charge. Since that time, the development of the electronic theory began, which explained such phenomena as thermionic emission, the appearance of cathode rays. The merit of creating the electronic theory belongs mainly to the Dutch physicist G.A. Lorentz, who in his work "Theory of Electrons" (1909) organically connected Maxwell's theory of the electromagnetic field with the electrical properties of matter, considered as a set of elementary electric charges.

Based on electronic representations in the first quarter of the 20th century. developed the theory of dielectrics and magnets. The theory of semiconductors is currently being developed. The study of electrical phenomena led to the modern theory of the structure of matter. The successes of physics in this direction culminated in the discovery of ways to release nuclear energy, which qualitatively raised the science and technology of mankind to a new stage of development.

It should be noted that in many technical applications electricity, in the doctrine of electricity and magnetism, the primacy belongs to Russian figures of science and technology. So, for example, Russian scientists and engineers invented and used for practice electroplating and electroplating, electric welding, electric lighting, electric motors, and radio. They developed many questions that are not only of great theoretical interest, but also of great practical importance. This includes the physics of dielectrics, semiconductors, magnets, gas discharge physics, thermionic emission, photoelectric effect, electromagnetic oscillations and radio waves, etc. recent times problems of direct conversion of solar energy into electrical energy, the creation of magnetohydrodynamic sources of electricity, "fuel cells". Russian scientists play a leading role in research aimed at solving the most important scientific and technical problem of our time - the problem of creating controlled thermonuclear reactions by using magnetic and electromagnetic fields for thermal insulation and heating of a highly ionized gas - plasma.

For a great contribution to the development of world science, Russian scientists - physicists I.E. Tammu, I.M. Frank and P.A. Cherenkov (1958), L.D. Landau (1962), N.G. Basov and A.M. Prokhorov (1964), P.L. Kapitsa (1978), Zh.I. Alferov (2000), V.L. Ginzburg and A.A. Abrikosov (2003) was awarded the Nobili Prizes.

Lecture 1. Electrostatics in vacuum
and substance. Electric field

The subject of classical electrodynamics. Electric charge and its discreteness. The theory of close action. Coulomb's law. Electric field strength. The principle of superposition of electric fields. The electric field of the dipole. Electrostatic field strength vector flow. The Ostrogradsky-Gauss theorem for an electric field in vacuum. The work of an electric field on the movement of an electric charge. Circulation of the electric field strength vector. The energy of an electric charge in an electric field. Potential and potential difference of the electric field. Electric field strength as a gradient of its potential. equipotential surfaces. Basic equations of electrostatics in vacuum. Some examples of electric fields generated by the simplest systems of electric charges.

The subject of classical electrodynamics

Classical electrodynamics is a theory that explains the behavior of an electromagnetic field that carries out electromagnetic interaction between electric charges.

The laws of classical macroscopic electrodynamics are formulated in Maxwell's equations, which allow you to determine the values ​​of the characteristics of the electromagnetic field - the electric field strength E and magnetic induction AT- in vacuum and in macroscopic bodies, depending on the distribution of electric charges and currents in space.

The interaction of stationary electric charges is described by the equations of electrostatics, which can be obtained as a consequence of Maxwell's equations.

The microscopic electromagnetic field created by individual charged particles in classical electrodynamics is determined by the Lorentz-Maxwell equations, which underlie the classical statistical theory of electromagnetic processes in macroscopic bodies. Averaging these equations leads to Maxwell's equations.

Among all known types of interaction, electromagnetic interaction ranks first in terms of breadth and variety of manifestations. This is due to the fact that all bodies are built of electrically charged (positive and negative) particles, the electromagnetic interaction between which, on the one hand, is many orders of magnitude more intense than the gravitational and weak one, and on the other hand, is long-range, in contrast to the strong interaction.

Electromagnetic interaction determines the structure of atomic shells, the adhesion of atoms into molecules (chemical bond forces) and the formation of condensed matter (interatomic interaction, intermolecular interaction).

The laws of classical electrodynamics are inapplicable at high frequencies and, accordingly, small lengths of electromagnetic waves, i.e. for processes occurring on small space-time intervals. In this case, the laws of quantum electrodynamics are valid.

1.2. Electric charge and its discreteness.
Short range theory

The development of physics has shown that the physical and chemical properties of a substance are largely determined by the forces of interaction due to the presence and interaction of electric charges of molecules and atoms of various substances.

It is known that in nature there are two types of electric charges: positive and negative. They can exist in the form of elementary particles: electrons, protons, positrons, positive and negative ions, etc., as well as "free electricity", but only in the form of electrons. Therefore, a positively charged body is a collection of electric charges with a lack of electrons, and a negatively charged body - with their excess. Charges of different signs compensate each other, therefore, in uncharged bodies there are always charges of both signs in such quantities that their total effect is compensated.

redistribution process positive and negative charges of uncharged bodies, or among separate parts of the same body, under the influence of various factors is called electrification.

Since the redistribution of free electrons occurs during electrification, for example, both interacting bodies are electrified, one of them being positive and the other negative. The number of charges (positive and negative) remains unchanged.

This implies the conclusion that charges are not created and do not disappear, but only redistributed between interacting bodies and parts of the same body, quantitatively remaining unchanged.

This is the meaning of the law of conservation of electric charges, which can be written mathematically as follows:

those. in an electrically isolated system, the algebraic sum of electric charges remains constant.

An electrically isolated system is understood as a system through which no other electric charges can penetrate.

It must be borne in mind that the total electric charge of an isolated system is relativistically invariant, since observers located in any given inertial coordinate system, measuring the charge, get the same value.

A number of experiments, in particular the laws of electrolysis, Millikan's experiment with a drop of oil, have shown that in nature electric charges are discrete to the charge of an electron. Any charge is a multiple of an integer number of the electron charge.

In the process of electrification, the charge changes discretely (quantized) by the value of the electron charge. Charge quantization is a universal law of nature.

In electrostatics, the properties and interactions of charges that are immobile in the frame of reference in which they are located are studied.

The presence of an electric charge in bodies causes them to interact with other charged bodies. At the same time, bodies charged with the same name repel each other, and charged oppositely, they attract.

Interaction in physics is understood as any influence of bodies or particles on each other, leading to a change in the state of their movement or to a change in their position in space. There are different types of interactions.

In Newtonian mechanics, the mutual action of bodies on each other is quantitatively characterized by force. A more general characteristic of interaction is potential energy.

Initially, in physics, the idea was established that the interaction between bodies can be carried out directly through empty space, which does not take part in the transfer of interaction. The transfer of interaction occurs instantly. Thus, it was believed that the movement of the Earth should immediately lead to a change in the gravitational force acting on the Moon. This was the meaning of the so-called theory of interaction, called the theory of long-range action. However, these ideas were abandoned as untrue after the discovery and study of the electromagnetic field.

It was proved that the interaction of electrically charged bodies is not instantaneous and the movement of one charged particle leads to a change in the forces acting on other particles, not at the same moment, but only after a finite time.

Each electrically charged particle creates an electromagnetic field that acts on other particles, i.e. interaction is transmitted through an "intermediary" - an electromagnetic field. The speed of propagation of an electromagnetic field is equal to the speed of propagation of light in a vacuum. A new theory of interaction arose - the theory of short-range interaction.

According to this theory, the interaction between bodies is carried out through certain fields (for example, gravitation through a gravitational field), continuously distributed in space.

After the advent of quantum field theory, the concept of interactions has changed significantly.

According to quantum theory, any field is not continuous, but has a discrete structure.

Owing to corpuscular-wave dualism, certain particles correspond to each field. Charged particles continuously emit and absorb photons, which form the electromagnetic field surrounding them. The electromagnetic interaction in quantum field theory is the result of the exchange of particles by photons (quanta) of the electromagnetic field, i.e. photons are carriers of such interaction. Similarly, other types of interactions arise as a result of the exchange of particles by quanta of the corresponding fields.

Despite the variety of influences of bodies on each other (depending on the interaction of their constituent elementary particles), in nature, according to modern data, there are only four types of fundamental interactions: gravitational, weak, electromagnetic and strong (in order of increasing interaction intensity). The intensities of interactions are determined by coupling constants (in particular, the electric charge for electromagnetic interaction is a coupling constant).

The modern quantum theory of electromagnetic interaction perfectly describes all known electromagnetic phenomena.

In the 60-70s of the century, a unified theory of weak and electromagnetic interactions (the so-called electroweak interaction) of leptons and quarks was basically built.

The modern theory of the strong interaction is quantum chromodynamics.

Attempts are being made to combine the electroweak and strong interactions into the so-called "Great Unification", as well as to include them in a single scheme of gravitational interaction.

ELECTRICITY

AND ELECTROMAGNETISM

Course of lectures in physics

for engineering students

specialties

ELECTROSTATICS

Lecture 1. Electric field in vacuum

Lecture plan

1.1. The subject of classical electrodynamics.

1.2. Electrostatics. Coulomb's law. Tension.

1.3. Gauss' theorem for an electrostatic field and its application to the calculation of electrostatic fields.

The subject of classical electrodynamics

Even in ancient times, experiments on electrification by friction were known (the term itself appeared later) and features of the force interaction of bodies after electrification (attraction and repulsion). It was found that there are only two types of electric charges, called conditionally positive and negative, and that charges of the same sign repel, unlike charges attract. To this (mostly qualitative) information, since the end of the eighteenth century, the revealed quantitative relationships and patterns that determine electrical phenomena began to be added.

It was found that the electric charge discrete, that is, the charge of any body is an integer multiple of elementary electric charge « e» ( e\u003d 1.6 10 19 C). Elementary particles: electron and proton are, respectively, carriers of elementary negative and positive charges. Generalization of the experimental data made it possible to formulate law of conservation of charge: the algebraic sum of the charges of any closed system (not exchanging charges with external bodies) remains unchanged. It turned out that electric charges invariant to coordinate transformations, i.e. do not depend on the reference system. The unit of electric charge in "SI" - 1 Coulomb (derived unit, defined in terms of current strength) - is the charge passing through the cross section of the conductor in one second at a current strength of 1A.

1.2. Electrostatics. Coulomb's law.
tension

In 1785, the French scientist C. Coulomb established the law of interaction of fixed point charges (whose dimensions are small compared to the distances to other charges): the force of interaction F between two point charges Q 1 , and Q 2 is proportional to the magnitudes of the charges and inversely proportional to the square of the distance between them.

, (1.1)

here – electrical constant; – medium permittivity- a dimensionless value showing how many times the force of interaction between charges in vacuum is weakened by this medium (for example: the dielectric constant of paraffin is 2; mica - 6, ethyl alcohol - 25; distilled water - 81; air - 1.0003 ≈ 1.0 ). The Coulomb force is directed along a straight line connecting the charges, that is, it is central and corresponds to attraction in the case of opposite charges and repulsion in the case of like charges.

In vector form, Coulomb's law has the form:

(1.1a)

If another charge is introduced into the space surrounding the electric charge, then the Coulomb force will act on it, that is, in the space around the charge there is force field. In this case, they are talking about electric field through which electric charges interact.

Consider the electric fields that are created by stationary charges and which are called electrostatic. If at some point BUT the field created by the charge Q, place charges alternately Q 1 ; Q 2 ;… Q n and determine the values ​​of the Coulomb force: , then according to (1.1) and, this is confirmed by experiment, the ratio . This value is taken as the power characteristic of the electrostatic field and is called tension

From (1.2) it follows that for Q\u003d 1, that is, the strength of the electrostatic field at a given point is determined by the force acting on a unit positive charge placed at this point of the field. In accordance with (1.1) and (1.2), the field strength of a point charge can be found by the formula

(1.3)

The direction of the vector coincides with the direction of the force acting on the positive charge. The dimension of tension in SI is .

In vector form:

Graphically, the electrostatic field is depicted using tension lines- lines, tangents to which at each point coincide with the direction of the vector at this point. Since at any given point in space the vector has only one direction, the lines of tension never intersect. So that with the help of lines of tension it was possible to characterize not only the direction, but also the magnitude of the strength of the electrostatic field, they are carried out with a certain density: the number of lines of tension dN penetrating a unit surface area dS, perpendicular to the lines of tension, must be equal to the numerical value of the vector . If we assign the dimension

E, then (1.4)

As an example on ( fig.1.1) is a graphic representation (using lines ) of electrostatic fields: a positive point charge (" a"); negative point charge (" b"); two point charges (" in") and the fields of two parallel planes uniformly charged with opposite charges (" G").

Fig.1.1

The electrostatic field is also characterized by a scalar quantity called tension vector flow through the surfaces under consideration F E. Elementary vector flow through the pad dS is entered as a scalar product according to the formula

(cm.. fig.1.2), here dS is the area of ​​the elementary area, is the unit vector of the normal to the area; is the angle between vectors and ; is the projection of the vector E onto the direction ; is a conditional vector whose modulus is equal to the area dS, and the direction is the same as " ".

Flow F E through the end surface S defined as

(1.6)

From expressions (1.5, 1.6) it follows that the sign F E depends on the sign of cos , which in turn depends on the relative position of the vectors and .

The direction is given by the arrangement of electric charges, and for the direction for a closed surface S is the direction of the normal coming out of the area covered by the closed surface S. Thus, the flow of the electrostatic field strength vector through the considered surface S is proportional to the number of lines of the vector , penetrating this surface.

Fig.1.2

Consider an electrostatic field created by a system of fixed point charges Q 1 ; Q 2 ;… Q n , at some point of which there is a charge Q. The experiment shows that for the Coulomb forces, the principle of independence of the action of forces acting in mechanics is valid - the resulting force acting from the side of the field on the charge Q, is equal to the vector sum of the forces applied to it by each of the charges Q i:

According to (1.2) , where is the strength of the resulting field; is the charge field strength Q i. Substituting these expressions into (1.7) we obtain the relation

expressing superposition principle(overlays) electrostatic fields: the field strength of a system of fixed point charges at some point is equal to the vector sum of the field strengths created at this point by each of the charges separately. The principle of superposition allows you to calculate the electrostatic fields of any system of fixed charges, since if the charges are not point charges, then they can always be reduced to a set of point charges.