The interaction of these two effects is main theme Newtonian mechanics.

Others important concepts This branch of physics is energy, momentum, angular momentum, which can be transferred between objects in the process of interaction. The energy of a mechanical system consists of its kinetic (energy of motion) and potential (depending on the position of the body relative to other bodies) energies. Fundamental conservation laws apply to these physical quantities.

1. History

Basics classical mechanics were laid down by Galileo, as well as Copernicus and Kepler when studying the laws of motion of celestial bodies, and for a long time mechanics and physics were considered in the context of describing astronomical events.

Ideas heliocentric system were further formalized by Kepler in his three laws of motion celestial bodies. In particular, Kepler's second law states that all planets in the solar system move in elliptical orbits, with the Sun as one of their focuses.

The next important contribution to the foundation of classical mechanics was made by Galileo, who, exploring the fundamental laws of mechanical motion of bodies, in particular under the influence of the forces of gravity, formulated five universal laws of motion.

But still, the laurels of the main founder of classical mechanics belong to Isaac Newton, who in his work “Mathematical Principles of Natural Philosophy” carried out a synthesis of those concepts in the physics of mechanical motion that were formulated by his predecessors. Newton formulated three fundamental laws of motion, which were named after him, as well as the law of universal gravitation, which drew a line under Galileo's investigations of the phenomenon free fall tel. Thus, a new picture of the world and its basic laws was created to replace the outdated Aristotelian one.

2. Limitations of classical mechanics

Classical mechanics gives accurate results for the systems we encounter in everyday life. But they become incorrect for systems whose speed approaches the speed of light, where it is replaced by relativistic mechanics, or for very small systems where the laws of quantum mechanics apply. For systems that combine both of these properties, relativistic quantum field theory is used instead of classical mechanics. For systems with very a large number components, or degrees of freedom, classical mechanics can also be adequate, but methods of statistical mechanics are used

Classical mechanics is widely used because, firstly, it is much simpler and easier to apply than the theories listed above, and, secondly, it has great opportunities for approximation and application to a very wide class physical objects, starting from the usual ones, such as a top or a ball, to great astronomical objects (planets, galaxies) and very microscopic ones (organic molecules).

3. Mathematical apparatus

Basic mathematics classical mechanics- differential and integral calculus, developed specifically for this by Newton and Leibniz. In its classical formulation, mechanics is based on Newton's three laws.

4. Statement of the basics of the theory

The following is a presentation of the basic concepts of classical mechanics. For simplicity, we will use the concept of a material point as an object whose dimensions can be neglected. Movement material point determined a small amount parameters: position, mass and forces applied to it.

In reality, the dimensions of every object that classical mechanics deals with are non-zero. A material point, such as an electron, obeys the laws quantum mechanics. Objects with non-zero dimensions have much more complex behavior, because they internal state can change - for example, a ball in motion can also rotate. Nevertheless, the results obtained for material points can be applied to such bodies if we consider them as a collection of many interacting material points. Such complex objects can behave like material points if their sizes are insignificant on the scale of a particular physical problem.

4.1. Position, radius vector and its derivatives

The position of an object (material point) is determined relative to a fixed point in space, which is called the origin. It can be specified by the coordinates of this point (for example, in the Cartesian coordinate system) or by a radius vector r, drawn from the origin to this point. In reality, a material point can move over time, so the radius vector is general case is a function of time. In classical mechanics, in contrast to relativistic mechanics, it is believed that the flow of time is the same in all reference systems.

4.1.1. Trajectory

A trajectory is the totality of all positions of a moving material point - in the general case, it is a curved line, the appearance of which depends on the nature of the point’s movement and the chosen reference system.

4.1.2. Moving

If all forces acting on a particle are conservative, and V is the total potential energy obtained by adding the potential energies of all forces, then

Those. total energy E = T + V persists over time. This is a manifestation of one of the fundamental physical laws of conservation. In classical mechanics it can be useful practically, because many types of forces in nature are conservative.

Mechanics- a branch of physics that studies the laws of changes in the positions of bodies in space over time and the causes that cause them, based on Newton’s laws. Therefore, it is often called “Newtonian mechanics”.

Classical mechanics is divided into:

static(which considers the balance of bodies)

kinematics(which studies geometric property movement without considering its reasons)

dynamics(which considers the movement of bodies).

Basic concepts of mechanics:

Space. It is believed that the movement of bodies occurs in space, which is Euclidean, absolute (independent of the observer), homogeneous (any two points in space are indistinguishable) and isotropic (any two directions in space are indistinguishable).

Time- fundamental concept, not defined in classical mechanics. It is believed that time is absolute, homogeneous and isotropic (the equations of classical mechanics do not depend on the direction of the flow of time)

Frame of reference– consists of a reference body (a certain body, real or imaginary, relative to which the movement of a mechanical system is considered) and a coordinate system

Material point- an object whose dimensions can be neglected in the problem. In fact, any body that obeys the laws of classical mechanics necessarily has a non-zero size. Non-zero size bodies can experience complex movements, since their internal configuration may change, for example, the body may rotate or deform. However, in certain cases To similar bodies The results obtained for material points are applicable if we consider such bodies as aggregates of a large number of interacting material points.

Weight- a measure of the inertia of bodies.

Radius vector- a vector drawn from the origin of coordinates to the point where the body is located characterizes the position of the body in space.

Speed is a characteristic of changes in body position over time, defined as the derivative of the path with respect to time.

Acceleration- speed speed changes, is defined as the derivative of speed with respect to time.

Pulse- vector physical quantity, equal to the product mass of a material point on its speed.

Kinetic energy- the energy of motion of a material point, defined as half the product of the mass of the body by the square of its speed.

Strength- a physical quantity characterizing the degree of interaction of bodies with each other. In fact, the definition of force is Newton's second law.

Conservative force- a force whose work does not depend on the shape of the trajectory (depends only on the initial and end point application of forces). Conservative forces are those forces whose work along any closed trajectory is equal to 0. If only conservative forces, That mechanical energy system is saved.

Dissipative forces- forces, under the action of which on mechanical system its total mechanical energy decreases (that is, dissipates), turning into other, non-mechanical forms of energy, for example, into heat.

Basic laws of mechanics

Galileo's principle of relativity- the basic principle on which classical mechanics is based is the principle of relativity, formulated on the basis empirical observations G. Galileo. According to this principle, there are infinitely many reference systems in which a free body is at rest or moving with a speed constant in magnitude and direction. These reference systems are called inertial and move relative to each other uniformly and rectilinearly. In all inertial systems reference, the properties of space and time are the same, and all processes in mechanical systems obey the same laws.

Newton's laws

The basis of classical mechanics is Newton's three laws.

Newton's first law establishes the presence of the property of inertia in material bodies and postulates the presence of such reference systems in which the movement of a free body occurs at a constant speed (such reference systems are called inertial).

Newton's second law introduces the concept of force as a measure of the interaction of a body and, on the basis of empirical facts, postulates a connection between the magnitude of the force, the acceleration of the body and its inertia (characterized by mass). In mathematical formulation, Newton's second law is most often written as follows:

Where F-resulting vector of forces acting on the body;

a- body acceleration vector;

m is body weight.

Newton's third law- for every force acting on the first body from the second, there is an opposing force, equal in magnitude and opposite in direction, acting on the second body from the first.

Law of Conservation of Energy

The law of conservation of energy is a consequence of Newton's laws for closed systems in which only conservative forces act. Total mechanical energy closed system bodies between which only conservative forces act remains constant.

Theory of machines and mechanisms

Basic concepts and definitions.

The theory of mechanisms and machines deals with the research and development of high-performance mechanisms and machines.

Mechanism- a set of moving material bodies, one of which is fixed, and all the others perform well-defined movements relative to the stationary material body.

Links – material bodies, of which the mechanism consists.

Rack- a fixed link.

The stand is depicted. The link to which movement is initially reported is called input(initial, leading). The link that makes the movement for which the mechanism is designed - day off link

Crank-slider mechanism

If this is a compressor, then link 1 is the input, and link 3 is the output.

If this is an internal combustion engine mechanism, then link 3 is the input, and link 1 is the output.

Kinematic pair- a movable connection of links that allows them relative motion. All kinematic pairs in the diagram are designated by letters Latin alphabet, for example A, B, C, etc.

If, then K.P. – rotational; if, then progressive.

Numbering order of links:

input link – 1;

stand is the last number.

The links are:

simple - consist of one piece;

complex - consist of several, rigidly fastened to each other and performing the same movement.

For example, the connecting rod group of an internal combustion engine mechanism.

The links, connecting to each other, form kinematic chains, which are divided into:

simple and complex;

closed and open.

Car– a technical device, as a result of the implementation of a certain kind of technological process, can automate or mechanize human labor.

Machines can be divided into types:

energy;

technological;

transport;

informational.

Energy machines are divided into:

engines;

transforming machines.

Engine- a technical device that converts one type of energy into another. For example, internal combustion engine.

Transformer machine- a technical device that consumes energy from the outside and performs useful work. For example, pumps, machines, presses.

Technical combination of engine and technological (working machine) – Machine unit(MA).

The engine has a certain mechanical characteristic, and so does the working machine.

 1 – speed at which the motor shaft rotates;

 2 – the speed at which the main shaft of the working machine will rotate.

 1 and  2 must be put in correspondence with each other.

For example, the speed n 1 =7000 rpm, and n 2 =70 rpm.

To harmonize the mechanical characteristics of the engine and the working machine, a transmission mechanism is installed between them, which has its own mechanical characteristics.

u P =1/2=700/70=10

The following can be used as a transmission mechanism:

friction transmissions (using friction);

chain transmissions (motorcycle drive);

gears.

Lever mechanisms are most often used as a working machine.

Main types of lever mechanisms.

1. Crank-slider mechanism.

a) central (Fig. 1);

b) off-axis (deoxyl) (Fig. 2);

e - eccentricity

Rice. 2

1-crank, because the link commits full turn around its axis;

2-connecting rod, not connected to the rack, makes a flat movement;

3-slider (piston), makes a translational movement;

2. Four-joint mechanism.

Links 1,3 can be cranks.

If gears 1 and 3 are cranks, then the mechanism is double-crank.

If star 1 is a crank (makes a full revolution), and star 3 is a rocker arm (makes an incomplete revolution), then the mechanism is a crank-rocker arm.

If the stars are 1.3 - rocker arms, then the mechanism is double-rocker.

3. Rocker mechanism.

1 - crank;

2 - the rocker stone (bushing) together with star 1 makes a full revolution around A (1 and 2 are the same), and also moves along star 3, causing it to rotate;

3 - rocker arm (scene).

4.Hydraulic cylinder

(kinematically similar to a rocker mechanism).

During the design process, the designer solves two problems:

analysis(explores ready mechanism);

synthesis(a new mechanism is being designed according to the required parameters);

Structural analysis of the mechanism.

Concepts about kinematic pairs and their classification.

Two links fixedly connected to each other form a kinematic pair. All kinematic pairs are subject to two independent classifications:

Examples of pair classification:

Let's consider the kinematic pair “screw-nut”. The number of degrees of mobility of this pair is 1, and the number of imposed connections is 5. This pair will be a fifth class pair; you can choose only one type of movement for a screw or nut freely, and the second movement will be accompanying.

Kinematic chain– links interconnected by kinematic pairs of different classes.

Kinematic chains can be spatial or flat.

Spatial kinematic chains– chains whose links move in different planes.

Flat kinematic chains– chains whose links move in the same or parallel planes.

Concepts about the degree of mobility of kinematic chains and mechanisms.

We denote the number of links freely floating in space as . For links, the degree of mobility can be determined by the formula:. We form a kinematic chain from these links by connecting the links together in pairs of different classes. The number of pairs of different classes is denoted by, where is class, that is: - the number of pairs of the first class, for which, a; - the number of pairs of the second class, for which, a; - the number of pairs of the third class, for which, a; - the number of pairs of the fourth class, for which, a; is the number of pairs of the fifth class, for which, a. The degree of mobility of the formed kinematic chain can be determined by the formula:.

We form a mechanism from a kinematic chain. One of the main features of the mechanism is the presence of a stand (body, base), around which the remaining links move under the action of the leading link (links).

The degree of mobility of the mechanism is usually denoted by . Let's turn one of the links of the kinematic chain into a stand, that is, take away all six degrees of mobility from it, then: - Somov-Malyshev formula.

In a flat system, the maximum number of degrees of freedom is two. Therefore, the degree of mobility of a planar kinetic chain can be determined by the following formula:. The degree of mobility of a flat mechanism is determined by the Chebyshev formula:, where is the number of moving links. Using the definition of higher and lower kinematic pairs, Chebyshev’s formula can be written as follows:

An example of determining the degree of mobility.

The top scientific creativity I. Newton is his immortal work “Mathematical Principles of Natural Philosophy,” first published in 1687. In it he summarized the results obtained by his predecessors and his own own research and created for the first time a single harmonious system of earthly and celestial mechanics, which formed the basis of all classical physics.

Here Newton gave definitions of the initial concepts - the amount of matter equivalent to mass, density; momentum equivalent to impulse, and various types strength. Formulating the concept of the amount of matter, he proceeded from the idea that atoms consist of some single primary matter; density was understood as the degree of filling a unit volume of a body with primary matter.

This work sets out Newton's doctrine of universal gravitation, on the basis of which he developed the theory of the motion of planets, satellites and comets that form the solar system. Based on this law, he explained the phenomenon of tides and the compression of Jupiter. Newton's concept was the basis for many technical achievements for a long time. Many methods were formed on its foundation scientific research V various areas natural sciences.

The result of the development of classical mechanics was the creation of a unified mechanical picture of the world, within the framework of which all the qualitative diversity of the world was explained by differences in the movement of bodies, subject to the laws of Newtonian mechanics.

Newton's mechanics, in contrast to previous mechanical concepts, made it possible to solve the problem of any stage of motion, both previous and subsequent, and at any point in space at known facts, causing this movement, as well as the inverse problem of determining the magnitude and direction of action of these factors at any point with the basic elements of movement known. Thanks to this, Newtonian mechanics could be used as a method quantitative analysis mechanical movement.

Law of Universal Gravitation.

Law universal gravity was discovered by I. Newton in 1682. According to his hypothesis, attractive forces act between all bodies of the Universe, directed along the line connecting the centers of mass. In a body in the form homogeneous ball the center of mass coincides with the center of the ball.

In subsequent years, Newton tried to find a physical explanation for the laws of planetary motion discovered by I. Kepler in early XVII century, and give a quantitative expression for gravitational forces. So, knowing how the planets move, Newton wanted to determine what forces act on them. This path is called inverse problem mechanics.

If the main task of mechanics is to determine the coordinates of a body known mass and its speed at any time according to known forces acting on the body, then when solving the inverse problem it is necessary to determine the forces acting on the body if it is known how it moves.

The solution to this problem led Newton to the discovery of the law of universal gravitation: “All bodies are attracted to each other with a force directly proportional to their masses and inversely proportional to the square of the distance between them.”

There are several important points to make regarding this law.

1, its action in explicit form applies to all physical material bodies in the Universe without exception.

2 the force of gravity of the Earth at its surface in equally affects all material bodies located at any point globe. There's a force on us right now gravity, and we really feel it as our weight. If we drop something, under the influence of the same force it will uniformly accelerate towards the ground.

The action of universal gravitational forces in nature explains many phenomena: the movement of planets in the solar system, artificial satellites Earth - they all find an explanation based on the law of universal gravitation and the laws of dynamics.

Newton was the first to suggest that gravitational forces determine not only the movement of planets solar system; they act between any bodies in the Universe. One of the manifestations of the force of universal gravitation is the force of gravity - this is the common name for the force of attraction of bodies towards the Earth near its surface.

The force of gravity is directed towards the center of the Earth. In the absence of other forces, the body falls freely to the Earth with the acceleration of gravity.

Three principles of mechanics.

Newton's laws of mechanics, three laws underlying the so-called. classical mechanics. Formulated by I. Newton (1687).

First Law: “Every body continues to be maintained in its state of rest or uniform and rectilinear movement until and unless it is forced by applied forces to change this state.”

Second law: “The change in momentum is proportional to the applied driving force and occurs in the direction of the straight line along which this force acts.”

Third law: “An action always has an equal and opposite reaction, otherwise, the interactions of two bodies on each other are equal and directed in opposite sides" N. z. m. appeared as a result of generalization of numerous observations, experiments and theoretical research G. Galileo, H. Huygens, Newton himself, etc.

According to modern ideas and terminology, in the first and second laws, a body should be understood as a material point, and motion should be understood as motion relative to an inertial reference system. Mathematical expression the second law in classical mechanics has the form or mw = F, where m is the mass of a point, u is its speed, and w is acceleration, F is the acting force.

N. z. m. cease to be valid for the movement of objects of very small sizes (elementary particles) and for movements at speeds close to the speed of light

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Mechanics is a part of physics that studies the patterns of mechanical movement and the reasons that cause or change this movement.

Mechanics, in turn, is divided into kinematics, dynamics and statics.

Mechanical movement is a change in the relative position of bodies or parts of the body over time.

Weight is a scalar physical quantity that quantitatively characterizes the inert and gravitational properties of matter.

Inertia- this is the desire of the body to maintain a state of rest or uniform rectilinear motion.

Inert mass characterizes the body’s ability to resist a change in its state (rest or motion), for example, in Newton’s second law

Gravitational mass characterizes the ability of a body to create a gravitational field, which is characterized by a vector quantity called tension. Tension gravitational field point mass is equal to:

Gravitational mass characterizes the ability of a body to interact with the gravitational field:

n equivalence principle gravitational and inertial masses: each mass is both inertial and gravitational.

The mass of a body depends on the density of the substance ρ and the size of the body (body volume V):

The concept of mass is not identical to the concepts of weight and gravity. It does not depend on gravitational fields and accelerations.

Moment of inertia– a tensor physical quantity that quantitatively characterizes the inertia of a solid body, manifested in rotational motion.

When describing rotational motion, specifying mass is not enough. The inertia of a body in rotational motion depends not only on mass, but also on its distribution relative to the axis of rotation.

1. Moment of inertia of a material point

where m is the mass of the material point; r – distance from the point to the axis of rotation.

2. Moment of inertia of a system of material points

3. The moment of inertia is absolutely solid

Strength is a vector physical quantity that is a measure mechanical impact on the body from other bodies or fields, as a result of which the body acquires acceleration or deforms (changes its shape or size).

Mechanics uses various models to describe mechanical motion.

Material point(m.t.) is a body with mass, the dimensions of which can be neglected in this problem.

Absolutely rigid body(a.t.t.) is a body that does not deform during movement, that is, the distance between any two points during movement remains unchanged.
§ 2. Laws of motion.

• 
  First Law n Newton : every material point (body) maintains a state of rest or uniform rectilinear motion until the influence of other bodies forces it to change this state.

• 
  Newton's second law (basic law of dynamics forward motion): the rate of change of momentum of a material point (body) is equal to the sum of the forces acting on it

• 
  Newton's third law : every action of material points (bodies) on each other is in the nature of interaction; the forces with which material points act on each other are always equal in magnitude, oppositely directed and act along the straight line connecting these points

here is the force acting on the first material point from the second; – force acting on the second material point from the first. These forces are applied to different material points (bodies), always act in pairs and are forces of the same nature.

here is the gravitational constant. .

Conservation laws in classical mechanics.

Conservation laws are satisfied in closed systems of interacting bodies.

A system is called closed if no external forces act on the system.

Pulse – vector physical quantity that quantitatively characterizes the reserve of translational motion:

Law of conservation of momentum systems of material points(m.t.): in closed systems m.t. full momentum is conserved

Where - speed i-th material point before interaction; – its speed after interaction.

Momentum – physical vector quantity that quantitatively characterizes the reserve of rotational motion.

– momentum of the material point, – radius vector of the material point.
Law of conservation of angular momentum : in a closed system the total angular momentum is conserved:

A physical quantity that characterizes the ability of a body or system of bodies to do work is called energy.

Energy – scalar physical quantity, which is the most general characteristic system state.

The state of a system is determined by its motion and configuration, i.e. relative position its parts. The motion of the system is characterized by kinetic energy K, and the configuration (the presence of a body in a potential field of forces) is characterized by potential energy U.

Total Energy is defined as the sum:

E = K + U + E internal,

where E internal – internal energy bodies.

Kinetic and potential energy add up to mechanical energy .

Einstein's formula(relationship between energy and mass):

In the reference system associated with the center of mass of the m.t. system, m = m 0 is the rest mass, and E = E 0 = m 0 . c 2 – rest energy.

Internal energy is determined in the reference system associated with the body itself, that is, the internal energy is at the same time the rest energy.

Kinetic energy – this is the energy of mechanical movement of a body or system of bodies. Relativistic kinetic energy determined by the formula

At low speeds v
.

Potential energy – a scalar physical quantity characterizing the interaction of bodies with other bodies or with fields.

Examples:

potential energy of elastic interaction

• 
  potential energy gravitational interaction point masses

Law of Conservation of Energy : the total energy of a closed system of material points is conserved

In the absence of energy dissipation (dissipation), both total and mechanical energies are conserved. IN dissipative systems total energy is conserved, but mechanical energy is not conserved.

Source electromagnetic field is an electric charge.

Electric charge - this is a property of some elementary particles enter into electromagnetic interaction.

Properties of electric charge :

1. Electric charge can be positive and negative (it is generally accepted that a proton is positively charged and an electron is negatively charged).

2. Electric charge is quantized. Quantum of electric charge – elementary electric charge (e = 1.610 –19 C). In a free state, all charges are multiples of an integer number of elementary electrical charges:

3. Law of conservation of charge: the total electric charge of a closed system is conserved in all processes occurring with the participation of charged particles:

q 1 + q 2 +...+ q N = q 1 * + q 2 * +...+ q N * .

4. relativistic invariance: the value of the total charge of the system does not depend on the movement of charge carriers (the charge of moving and resting particles is the same). In other words, in all ISOs the amount of charge of any particle or body is the same.

Description of the electromagnetic field.

Charges interact with each other (Fig. 1). The magnitude of the force with which charges of the same sign repel each other, and charges different sign attract each other, determined empirically established law Pendant:

Here, is the electric constant.

What is the mechanism of interaction between charged bodies? We can put forward the following hypothesis: bodies with electric charge, generate an electromagnetic field. In turn, the electromagnetic field affects other charged bodies located in this field. A new one has arisen material object– electromagnetic field.

Experience shows that in any electromagnetic field a force acts on a stationary charge, the magnitude of which depends only on the magnitude of the charge (the magnitude of the force is proportional to the magnitude of the charge) and its position in the field. Each point in the field can be associated with a certain vector, which is the coefficient of proportionality between the force acting on a stationary charge in the field and the charge. Then the force with which the field acts on a stationary charge can be determined by the formula:

The force exerted by the electromagnetic field on a stationary charge is called electrical force. Vector quantity, characterizing the state of the field that determines the action, is called electrical intensity electromagnetic field.

Further experiments with charges show that the vector does not completely characterize the electromagnetic field. If the charge begins to move, then some additional force appears, the magnitude and direction of which are in no way related to the magnitude and direction of the vector. The additional force that occurs when a charge moves in an electromagnetic field is called magnetic force. Experience shows that magnetic force depends on the charge and on the magnitude and direction of the velocity vector. If you move a test charge through any fixed point of the field with the same speed, but in different directions, then the magnetic force will be different each time. However, always. Further analysis of the experimental facts made it possible to establish that for each point of the electromagnetic field there is a single direction MN (Fig. 2), which has the following properties:

If a certain vector is directed along the direction MN, which has the meaning of a coefficient of proportionality between the magnetic force and the product, then the assignment , and unambiguously characterizes the state of the field that causes the appearance of . The vector was called a vector electromagnetic induction. Since and , then

In an electromagnetic field, a charge moving at speed q is acted upon by the electromagnetic Lorentz force (Fig. 3):

.
Vectors and , that is, six numbers, are equal components of a single electromagnetic field (components of the electromagnetic field tensor). In a particular case, it may turn out that everything or all ; then the electromagnetic field is reduced to either electric or magnetic fields.

The experiment confirmed the correctness of the constructed two-vector model of the electromagnetic field. In this model, each point of the electromagnetic field is given a pair of vectors and . The model we have constructed is a model of a continuous field, since the functions and , which describe the field, are continuous functions coordinates

Theory electromagnetic phenomena, using a continuous field model, is called classical.

In reality, the field, like matter, is discrete. But this begins to affect only distances comparable to the sizes of elementary particles. The discreteness of the electromagnetic field is taken into account in quantum theory.

Superposition principle.

Fields are usually depicted using power lines.

power line is a line whose tangent at each point coincides with the field strength vector.

D
for point stationary charges picture of power lines electrostatic field shown in Fig. 6.

The vector of the intensity of the electrostatic field created by a point charge is determined by the formula (Fig. 7 a and b) the magnetic field line is constructed so that at each point of the field line the vector is directed tangentially to this line. The magnetic field lines are closed (Fig. 8). This suggests that the magnetic field is a vortex field.

And if the field creates not one, but several point charges? Do the charges influence each other or does each charge in the system contribute to the resulting field independently of the others? Will the electromagnetic field created i-th charge in the absence of other charges the same as the field created by i-th charge in the presence of other charges?

Superposition principle : electromagnetic field arbitrary system charges are the result of the addition of fields that would be created by each of the elementary charges of this system in the absence of the others:

And .
Laws of the electromagnetic field

The laws of the electromagnetic field are formulated in the form of a system of Maxwell's equations.

First

From Maxwell's first equation it follows that electrostatic field is potential (converging or diverging) and its source is stationary electric charges.

Second Maxwell's equation for the magnetostatic field:

From Maxwell's second equation it follows that The magnetostatic field is vortex, not potential, and has no point sources.

Third Maxwell's equation for the electrostatic field:

From Maxwell's third equation it follows that the electrostatic field is not vortex.

In electrodynamics (for an alternating electromagnetic field), Maxwell's third equation is:

i.e. the electric field is not potential (not Coulomb), but vortex and is created by the alternating flux of the magnetic field induction vector.

Fourth Maxwell's equation for the magnetostatic field

From fourth equation Maxwell in magnetostatics it follows that magnetic field is vortex and is created by constant electric currents or moving charges. The direction of twist of the magnetic field lines is determined by the rule of the right screw (Fig. 9).

R
is.9

In electrodynamics, Maxwell's fourth equation is:

The first term in this equation is the conduction current I, associated with the movement of charges and creating a magnetic field.

The second term in this equation is the “displacement current in vacuum”, i.e. the variable flux of the tension vector electric field.

The main provisions and conclusions of Maxwell's theory are as follows.

A change in electric field over time leads to the appearance of a magnetic field and vice versa. Therefore, electromagnetic waves exist.

Electromagnetic energy transfer occurs at a finite speed . Baud rate electromagnetic vibrations equal to the speed of light. From this followed the fundamental identity of electromagnetic and optical phenomena.

Mechanics is a branch of physics that studies one of the simplest and most general forms movement in nature, called mechanical movement.

Mechanical movement consists in changing the position of bodies or their parts relative to each other over time. Thus, mechanical motion is performed by planets revolving in closed orbits around the Sun; different bodies, moving along the surface of the Earth; electrons moving under the influence of an electromagnetic field, etc. Mechanical movement is present in other more complex forms matter as an integral, but not exhaustive part.

Depending on the nature of the objects being studied, mechanics is divided into the mechanics of a material point, the mechanics of a solid body and the mechanics of a continuous medium.

The principles of mechanics were first formulated by I. Newton (1687) on the basis of an experimental study of the motion of macrobodies with small velocities compared to the speed of light in a vacuum (3·10 8 m/s).

Macrobodies are called ordinary bodies that surround us, that is, bodies consisting of a huge number of molecules and atoms.

Mechanics, which studies the movement of macrobodies at speeds much lower than the speed of light in a vacuum, is called classical.

Classical mechanics is based on Newton’s following ideas about the properties of space and time.

Any physical process flows in space and time. This is evident from the fact that in all areas of physical phenomena, each law explicitly or implicitly contains space-time quantities - distances and time intervals.

Space, which has three dimensions, obeys Euclidean geometry, that is, it is flat.

Distances are measured by scales, the main property of which is that two scales that once coincide in length always remain equal to each other, that is, they coincide with each subsequent overlap.

Time intervals are measured in hours, and the role of the latter can be performed by any system that performs a repeating process.

The main feature of the ideas of classical mechanics about the sizes of bodies and time intervals is their absoluteness: the scale always has the same length, no matter how it moves relative to the observer; two clocks that have the same speed and are once brought into line with each other show the same time regardless of how they move.

Space and time have remarkable properties symmetry, imposing restrictions on the occurrence of certain processes in them. These properties have been established experimentally and seem so obvious at first glance that there seems to be no need to single them out and deal with them. Meanwhile, if there were no spatial and temporal symmetry, no physical science could neither arise nor develop.

It turns out that space homogeneously And isotropically, and time - homogeneously.

The homogeneity of space consists in the fact that the same physical phenomena under the same conditions are performed in the same way various parts space. All points in space are thus completely indistinguishable, equal in rights, and any of them can be taken as the origin of the coordinate system. The homogeneity of space is manifested in the law of conservation of momentum.

Space also has isotropy: the same properties in all directions. The isotropy of space is manifested in the law of conservation of angular momentum.

The homogeneity of time lies in the fact that all moments of time are also equal, equivalent, that is, the occurrence of identical phenomena in the same conditions is the same, regardless of the time of their implementation and observation.

The uniformity of time is manifested in the law of conservation of energy.

Without these homogeneity properties, installed in Minsk physical law would be unfair in Moscow, and open today in the same place could be unfair tomorrow.

Classical mechanics recognizes the validity of the Galileo-Newton law of inertia, according to which a body, not subject to the influence of other bodies, moves rectilinearly and uniformly. This law asserts the existence of inertial frames of reference in which Newton's laws (as well as Galileo's principle of relativity) are satisfied. Galileo's principle of relativity states that all inertial frames of reference are mechanically equivalent to each other, all the laws of mechanics are the same in these reference frames, or, in other words, are invariant under Galilean transformations expressing the spatio-temporal relationship of any event in different inertial reference frames. Galileo's transformations show that the coordinates of any event are relative, that is, they have different meanings V different systems countdown; the moments in time when the event occurred are the same in different systems. The latter means that time flows in the same way in different reference systems. This circumstance seemed so obvious that it was not even stated as a special postulate.

In classical mechanics, the principle of long-range action is observed: the interactions of bodies propagate instantly, that is, with an infinitely high speed.

Depending on the speeds at which bodies move and the dimensions of the bodies themselves, mechanics is divided into classical, relativistic, and quantum.

As already indicated, the laws classical mechanics applicable only to the movement of macrobodies, the mass of which is much more mass atom, with low speeds compared to the speed of light in a vacuum.

Relativistic mechanics considers the movement of macrobodies at speeds close to the speed of light in a vacuum.

Quantum mechanics- mechanics of microparticles moving at speeds much lower than the speed of light in a vacuum.

Relativistic quantum mechanics - the mechanics of microparticles moving at speeds approaching the speed of light in a vacuum.

To determine whether a particle belongs to macroscopic ones, whether classic formulas, you need to use Heisenberg's uncertainty principle. According to quantum mechanics, real particles can be characterized by position and momentum only with some accuracy. The limit of this accuracy is determined as follows

Where
ΔX - coordinate uncertainty;
ΔP x - uncertainty of projection onto the momentum axis;
h is Planck’s constant equal to 1.05·10 -34 J·s;
"≥" - greater than the value, about...

Replacing momentum with the product of mass and velocity, we can write

From the formula it is clear that the smaller the mass of the particle, the less certain its coordinates and speed become. For macroscopic bodies, the practical applicability of the classical method of describing motion is beyond doubt. Let's say, for example, that we're talking about about the movement of a ball with a mass of 1 g. Usually the position of the ball can be practically determined with an accuracy of a tenth or a hundredth of a millimeter. In any case, it hardly makes sense to talk about an error in determining the position of a ball that is smaller than the size of an atom. Let us therefore put ΔX=10 -10 m. Then from the uncertainty relation we find

The simultaneous smallness of the values ​​of ΔX and ΔV x is proof of the practical applicability of the classical method of describing the motion of macrobodies.

Let's consider the movement of an electron in a hydrogen atom. The electron mass is 9.1·10 -31 kg. The error in the position of the electron ΔX in any case should not exceed the size of the atom, that is, ΔX<10 -10 м. Но тогда из соотношения неопределенностей получаем

This value is even greater than the speed of an electron in an atom, which is an order of magnitude equal to 10 6 m/s. In this situation, the classical picture of movement loses all meaning.

Mechanics are divided into kinematics, statics and dynamics. Kinematics describes the movement of bodies without being interested in the reasons that determined this movement; statics considers the conditions of equilibrium of bodies; dynamics studies the movement of bodies in connection with those reasons (interactions between bodies) that determine this or that nature of movement.

The real movements of bodies are so complex that when studying them, it is necessary to abstract from details that are unimportant for the movement under consideration (otherwise the problem would become so complicated that it would be practically impossible to solve it). For this purpose, concepts (abstractions, idealizations) are used, the applicability of which depends on the specific nature of the problem we are interested in, as well as on the degree of accuracy with which we want to obtain the result. Among these concepts, an important role is played by the concepts material point, system of material points, absolutely rigid body.

A material point is a physical concept with the help of which the translational motion of a body is described, if only its linear dimensions are small in comparison with the linear dimensions of other bodies within the given accuracy of determining the coordinates of the body, and the mass of the body is assigned to it.

In nature, material points do not exist. One and the same body, depending on the conditions, can be considered either as a material point or as a body of finite dimensions. Thus, the Earth moving around the Sun can be considered a material point. But when studying the rotation of the Earth around its axis, it can no longer be considered a material point, since the nature of this movement is significantly influenced by the shape and size of the Earth, and the path traversed by any point on the earth’s surface in a time equal to the period of its revolution around its axis is comparable with the linear dimensions of the globe. An airplane can be considered as a material point if we study the movement of its center of mass. But if it is necessary to take into account the influence of the environment or determine the forces in individual parts of the aircraft, then we must consider the aircraft as an absolutely rigid body.

An absolutely rigid body is a body whose deformations can be neglected under the conditions of a given problem.

A system of material points is a collection of bodies under consideration that represent material points.

The study of the motion of an arbitrary system of bodies comes down to the study of a system of interacting material points. It is natural, therefore, to begin the study of classical mechanics with the mechanics of one material point, and then move on to the study of a system of material points.