« Physics - Grade 10 "

What is the intermediary that carries out the interaction of charges?
How to determine which of the two fields is stronger? Suggest ways to compare fields.

Electric field strength.

The electric field is detected by the forces acting on the charge. It can be argued that we know everything we need about the field if we know the force acting on any charge at any point in the field. Therefore, it is necessary to introduce such a characteristic of the field, the knowledge of which will allow us to determine this force.

If we alternately place small charged bodies at the same point of the field and measure the forces, it will be found that the force acting on the charge from the field is directly proportional to this charge. Indeed, let the field be created by a point charge q 1 . According to Coulomb's law (14.2), a force proportional to the charge q acts on a point charge q. Therefore, the ratio of the force acting on a charge placed at a given point of the field to this charge for each point of the field does not depend on the charge and can be considered as a characteristic of the field.

The ratio of the force acting on a point charge placed at a given point in the field to this charge is called electric field strength.

Like a force, field strength - vector quantity; it is denoted by the letter:

Hence, the force acting on the charge q from the electric field is equal to:

Q. (14.8)

The direction of the vector is the same as the direction of the force acting on the positive charge and opposite to the direction of the force acting on the negative charge.

The unit of tension in SI is N/Cl.

Force lines of the electric field.

The electric field does not affect the sense organs. We do not see him. However, we can get some idea of ​​the distribution of the field if we draw the field strength vectors at several points in space (Fig. 14.9, a). The picture will be more visual if you draw continuous lines.

The lines, the tangent at each point of which coincides with the electric field strength vector, are called lines of force or field strength lines(Fig. 14.9, b).

The direction of the field lines allows you to determine the direction of the field strength vector at various points in the field, and the density (the number of lines per unit area) of the field lines shows where the field strength is greater. So, in Figures 14 10-14.13, the density of field lines at points A is greater than at points B. It is obvious that A > B.

One should not think that lines of tension actually exist like stretched elastic threads or cords, as Faraday himself assumed. The lines of tension only help visualize the distribution of the field in space. They are no more real than the meridians and parallels on the globe.

Field lines can be made visible. If the elongated crystals of an insulator (for example, quinine) are well mixed in a viscous liquid (for example, in castor oil) and charged bodies are placed there, then near these bodies the crystals will line up in chains along the lines of tension.

The figures show examples of tension lines: a positively charged ball (see Fig. 14.10), two oppositely charged balls (see Fig. 14.11), two like-charged balls (see Fig. 14.12), two plates whose charges are equal in absolute value and opposite in sign (see Fig. 14.13). The last example is especially important.

Figure 14.13 shows that in the space between the plates, the lines of force are basically parallel and at equal distances from each other: the electric field here is the same at all points.

An electric field whose intensity is the same at all points is called homogeneous.

In a limited area of ​​space, an electric field can be considered approximately uniform if the field strength inside this area changes insignificantly.

The lines of force of the electric field are not closed, they start on positive charges and end on negative ones. The lines of force are continuous and do not intersect, since the intersection would mean the absence of a certain direction of the electric field strength at a given point.

In the space surrounding the charge that is the source, is directly proportional to the amount of this charge and inversely to the square of the distance from this charge. The direction of the electric field according to the accepted rules is always from a positive charge towards a negative charge. This can be represented as if a test charge is placed in the space region of the electric field of the source and this test charge will either repel or attract (depending on the sign of the charge). The electric field is characterized by strength , which, being a vector quantity, can be represented graphically as an arrow having a length and direction. Anywhere the direction of the arrow indicates the direction of the electric field strength E, or simply - the direction of the field, and the length of the arrow is proportional to the numerical value of the electric field strength in this place. The farther the region of space is from the source of the field (charge Q), the smaller the length of the intensity vector. Moreover, the length of the vector decreases with distance to n times from some place in n 2 times, that is, inversely proportional to the square.

A more useful means of visualizing the vector nature of an electric field is to use such a concept as, or simply, lines of force. Instead of depicting countless vector arrows in space surrounding the source charge, it turned out to be useful to combine them into lines, where the vectors themselves are tangent to points on such lines.

As a result, successfully used to represent the vector picture of the electric field electric field lines, which come out of positive charges and into negative charges, and also extend to infinity in space. This representation allows you to see with the mind the electric field invisible to the human eye. However, such a representation is also convenient for gravitational forces and any other contactless long-range interactions.

The model of electric field lines includes an infinite number of them, but too high a density of the image of field lines reduces the ability to read field patterns, so their number is limited by readability.

Rules for drawing electric field lines

There are many rules for compiling such models of electrical power lines. All these rules are designed to provide the most information when visualizing (drawing) an electric field. One way is to depict field lines. One of the most common ways is to surround more charged objects with more lines, that is, a greater density of lines. Objects with a large charge create stronger electric fields and therefore the density (density) of lines around them is greater. The closer to the charge the source, the higher the density of field lines, and the greater the charge, the thicker the lines around it.

The second rule for drawing electric field lines involves drawing lines of a different type, such as those that intersect the first lines of force. perpendicular. This type of line is called equipotential lines, and in the case of a volumetric representation, one should speak of equipotential surfaces. This type of line forms closed contours and each point on such an equipotential line has the same value of the field potential. When any charged particle crosses such perpendicular lines of force lines (surfaces), then they talk about the work done by the charge. If the charge moves along equipotential lines (surfaces), then although it moves, no work is done. A charged particle, once in the electric field of another charge, begins to move, but in static electricity only stationary charges are considered. The movement of charges is called electric current, and work can be done by the charge carrier.

It is important to remember that electric field lines do not intersect, and lines of another type - equipotential, form closed loops. In the place where there is an intersection of two types of lines, the tangents to these lines are mutually perpendicular. Thus, something like a curved coordinate grid, or grating, is obtained, the cells of which, as well as the points of intersection of lines of different types, characterize the electric field.

Dashed lines are equipotential. Lines with arrows - electric field lines

Electric field consisting of two or more charges

For solitary individual charges electric field lines represent radial rays emerging from charges and going to infinity. What will be the configuration of field lines for two or more charges? To perform such a pattern, it must be remembered that we are dealing with a vector field, that is, with electric field strength vectors. To depict the field pattern, we need to perform the addition of the intensity vectors from two or more charges. The resulting vectors will represent the total field of several charges. How can lines of force be drawn in this case? It is important to remember that each point on the field line is single point contact with the electric field strength vector. This follows from the definition of a tangent in geometry. If from the beginning of each vector we construct a perpendicular in the form of long lines, then the mutual intersection of many such lines will depict the very desired line of force.

For a more accurate mathematical algebraic representation of the lines of force, it is necessary to compose the equations of the lines of force, and the vectors in this case will represent the first derivatives, the lines of the first order, which are the tangents. Such a task is sometimes extremely complex and requires computer calculations.

First of all, it is important to remember that the electric field from many charges is represented by the sum of the intensity vectors from each charge source. This is the basis to perform the construction of field lines in order to visualize the electric field.

Each charge introduced into the electric field leads to a change, even if insignificant, in the pattern of field lines. Such images are sometimes very attractive.

Electric field lines as a way to help the mind see reality

The concept of an electric field arose when scientists tried to explain the long-range action that occurs between charged objects. The concept of an electric field was first introduced by the 19th century physicist Michael Faraday. It was the result of Michael Faraday's perception invisible reality in the form of a picture of lines of force characterizing long-range action. Faraday did not think within the framework of one charge, but went further and expanded the boundaries of the mind. He suggested that a charged object (or mass in the case of gravity) affects space and introduced the concept of a field of such influence. Considering such fields, he was able to explain the behavior of charges and thereby revealed many of the secrets of electricity.

There are scalar and vector fields (in our case, the vector field will be electric). Accordingly, they are modeled by scalar or vector functions of coordinates, as well as time.

The scalar field is described by a function of the form φ. Such fields can be visualized using surfaces of the same level: φ (x, y, z) = c, c = const.

Let us define a vector that is directed towards the maximum growth of the function φ.

The absolute value of this vector determines the rate of change of the function φ.

Obviously, a scalar field generates a vector field.

Such an electric field is called potential, and the function φ is called potential. Surfaces of the same level are called equipotential surfaces. For example, consider an electric field.

For a visual display of the fields, the so-called electric field lines are built. They are also called vector lines. These are lines whose tangent at a point indicates the direction of the electric field. The number of lines that pass through the unit surface is proportional to the absolute value of the vector.

Let us introduce the concept of a vector differential along some line l. This vector is directed tangentially to the line l and is equal in absolute value to the differential dl.

Let some electric field be given, which must be represented as field lines of force. In other words, let's define the coefficient of stretching (compression) k of the vector so that it coincides with the differential. Equating the components of the differential and the vector, we obtain a system of equations. After integration it is possible to construct the equation of force lines.

In vector analysis, there are operations that provide information about which electric field lines are present in a particular case. Let us introduce the concept of “vector flow” on the surface S. The formal definition of the flow Ф has the following form: the quantity is considered as the product of the usual differential ds by the unit vector of the normal to the surface s. The unit vector is chosen so that it defines the outer normal of the surface.

It is possible to draw an analogy between the concept of a field flow and a substance flow: a substance per unit time passes through a surface, which in turn is perpendicular to the direction of the field flow. If the lines of force go out of the surface S, then the flow is positive, and if they do not go out, then it is negative. In general, the flow can be estimated by the number of lines of force that come out of the surface. On the other hand, the magnitude of the flux is proportional to the number of field lines penetrating the surface element.

The divergence of the vector function is calculated at the point whose band is the volume ΔV. S is the surface covering the volume ΔV. The divergence operation makes it possible to characterize points in space for the presence of field sources in it. When the surface S is compressed to the point P, the electric field lines penetrating the surface will remain in the same quantity. If a point in space is not a source of the field (leakage or sink), then when the surface is compressed to this point, the sum of the lines of force, starting from a certain moment, is equal to zero (the number of lines entering the surface S is equal to the number of lines emanating from this surface).

The closed loop integral L in the definition of the rotor operation is called the circulation of electricity along the loop L. The rotor operation characterizes the field at a point in space. The direction of the rotor determines the magnitude of the closed field flow around a given point (the rotor characterizes the field vortex) and its direction. Based on the definition of the rotor, by simple transformations, it is possible to calculate the projections of the electricity vector in the Cartesian coordinate system, as well as the electric field lines.

Electric charge (amount of electricity) is a physical scalar quantity that determines the ability of bodies to be a source of electromagnetic fields and take part in electromagnetic interaction. Electric charge was first introduced in Coulomb's law in 1785.

The unit of charge in the International System of Units (SI) is the pendant - an electric charge passing through the cross section of a conductor at a current of 1 A in a time of 1 s. A charge of one pendant is very large. If two charge carriers ( q 1 = q 2 = 1 C) placed in a vacuum at a distance of 1 m, then they would interact with a force of 9 10 9 H, that is, with the force with which the Earth's gravity would attract an object with a mass of about 1 million tons. The electric charge of a closed system is preserved in time and quantized - it changes in portions that are multiples of the elementary electric charge, that is, in other words, the algebraic sum of the electric charges of bodies or particles that form an electrically isolated system does not change during any processes occurring in this system.

Charge interaction The simplest and most everyday phenomenon in which the fact of the existence of electric charges in nature is revealed is the electrification of bodies upon contact. The ability of electric charges to both mutual attraction and mutual repulsion is explained by the existence of two different types of charges. One kind of electric charge is called positive, and the other is called negative. Oppositely charged bodies attract each other, and similarly charged bodies repel each other.

When two electrically neutral bodies come into contact, as a result of friction, charges pass from one body to another. In each of them, the equality of the sum of positive and negative charges is violated, and the bodies are charged differently.

When a body is electrified through influence, the uniform distribution of charges is disturbed in it. They are redistributed so that in one part of the body there is an excess of positive charges, and in another - negative. If these two parts are separated, then they will be charged differently.

The law of conservation of email. charge In the system under consideration, new electrically charged particles can form, for example, electrons - due to the phenomenon of ionization of atoms or molecules, ions - due to the phenomenon of electrolytic dissociation, etc. However, if the system is electrically isolated, then the algebraic sum of the charges of all particles, including again appearing in such a system is always equal to zero.

The law of conservation of electric charge is one of the fundamental laws of physics. It was first experimentally confirmed in 1843 by the English scientist Michael Faraday and is currently considered one of the fundamental laws of conservation in physics (similar to the laws of conservation of momentum and energy). Increasingly sensitive experimental tests of the law of conservation of charge, which continue to this day, have not yet revealed deviations from this law.

. Electric charge and its discreteness. The law of conservation of charge. The law of conservation of electric charge states that the algebraic sum of the charges of an electrically closed system is conserved. q, Q, e are designations of electric charge. Units of charge in SI [q]=Cl (Coulomb). 1mC = 10-3 C; 1 µC = 10-6 C; 1nC = 10-9 C; e = 1.6∙10-19 C is the elementary charge. The elementary charge, e is the minimum charge found in nature. Electron: qe = - e - electron charge; m = 9.1∙10-31 kg is the mass of the electron and positron. Positron, proton: qp = + e is the charge of the positron and proton. Any charged body contains an integer number of elementary charges: q = ± Ne; (1) Formula (1) expresses the discreteness principle of electric charge, where N = 1,2,3… is a positive integer. The law of conservation of electric charge: the charge of an electrically isolated system does not change over time: q = const. Coulomb's Law- one of the basic laws of electrostatics, which determines the force of interaction between two point electric charges.

The law was established in 1785 by Sh. Coulomb with the help of the torsion scales invented by him. Coulomb was interested not so much in electricity as in the manufacture of appliances. Having invented an extremely sensitive device for measuring force - a torsion balance, he was looking for ways to use it.

For suspension, the Pendant used a silk thread 10 cm long, which rotated 1 ° at a force of 3 * 10 -9 gf. With the help of this device, he established that the force of interaction between two electric charges and between two poles of magnets is inversely proportional to the square of the distance between the charges or poles.

Two point charges interact with each other in a vacuum with a force F , the value of which is proportional to the product of charges e 1 and e 2 and inversely proportional to the square of the distance r between them:

Proportionality factor k depends on the choice of the system of units of measurement (in the system of Gaussian units k= 1, in SI

ε 0 is the electrical constant).

Force F is directed along a straight line connecting the charges, and corresponds to attraction for unlike charges and repulsion for like charges.

If the interacting charges are in a homogeneous dielectric, with permittivity ε , then the force of interaction decreases in ε once:

Coulomb's law is also called the law that determines the strength of the interaction of two magnetic poles:

where m 1 and m 2 - magnetic charges,

μ is the magnetic permeability of the medium,

f is the coefficient of proportionality, depending on the choice of the system of units.

Electric field- a separate form of manifestation (along with the magnetic field) of the electromagnetic field.

During the development of physics, there were two approaches to explaining the causes of the interaction of electric charges.

According to the first version, the force action between separate charged bodies was explained by the presence of intermediate links that transmit this action, i.e. the presence of the environment surrounding the body, in which the action is transmitted from point to point with a finite speed. This theory is called short range theory .

According to the second version, the action is transmitted instantly over any distance, while the intermediate medium may be completely absent. One charge instantly "feels" the presence of another, while no changes occur in the surrounding space. This theory has been called long-range theory .

The concept of "electric field" was introduced by M. Faraday in the 30s of the XIX century.

According to Faraday, each charge at rest creates an electric field in the surrounding space. The field of one charge acts on another charge and vice versa (the concept of short-range action).

An electric field created by stationary charges that does not change with time is called electrostatic. The electrostatic field characterizes the interaction of fixed charges.

Electric field strength- a vector physical quantity characterizing the electric field at a given point and numerically equal to the ratio of the force acting on a fixed point charge placed at a given point of the field to the value of this charge:

This definition shows why the strength of the electric field is sometimes called the power characteristic of the electric field (indeed, the difference from the vector of the force acting on a charged particle is only in a constant factor).

At each point in space at a given moment of time there is its own value of the vector (generally speaking, it is different at different points in space), so this is a vector field. Formally, this is expressed in the notation

representing the electric field strength as a function of spatial coordinates (and time, since it can change over time). This field, together with the field of the magnetic induction vector, is an electromagnetic field, and the laws to which it obeys are the subject of electrodynamics.

The strength of an electric field in the International System of Units (SI) is measured in volts per meter [V/m] or in newtons per pendant [N/C].

The force with which an electromagnetic field acts on charged particles[

The total force with which an electromagnetic field (generally including electric and magnetic components) acts on a charged particle is expressed by the Lorentz force formula:

where q- the electric charge of the particle, - its speed, - the vector of magnetic induction (the main characteristic of the magnetic field), the oblique cross denotes the vector product. The formula is given in SI units.

The charges that create an electrostatic field can be distributed in space either discretely or continuously. In the first case, the field strength: n E = Σ Ei₃ i=t, where Ei is the strength at a certain point in the space of the field created by one i-th charge of the system, and n is the total number of discreet charges that are part of the system. An example of solving a problem based on the principle of superposition of electric fields. So, to determine the intensity of the electrostatic field, which is created in vacuum by stationary point charges q₁, q₂, …, qn, we use the formula: n E = (1/4πε₀) Σ (qi/r³i)ri i=t, where ri is the radius vector drawn from the point charge qi to the considered point of the field. Let's take another example. Determination of the strength of the electrostatic field, which is created in vacuum by an electric dipole. An electric dipole is a system of two equal in absolute value and, at the same time, opposite in sign charges q>0 and –q, the distance I between which is relatively small compared to the distance of the points under consideration. The arm of the dipole will be called the vector l, which is directed along the axis of the dipole to the positive charge from the negative one and is numerically equal to the distance I between them. The vector pₑ = ql is the electric moment of the dipole.

The strength E of the dipole field at any point: E = E₊ + E₋, where E₊ and E₋ are the field strengths of electric charges q and –q. Thus, at point A, which is located on the dipole axis, the dipole field strength in vacuum will be equal to E = (1/4πε₀)(2pₑ/r³) At point B, which is located on the perpendicular restored to the dipole axis from its middle: E = (1/4πε₀)(pₑ/r³) At an arbitrary point M sufficiently remote from the dipole (r≥l), the module of its field strength is The principle of superposition of electric fields consists of two statements: The Coulomb force of the interaction of two charges does not depend on the presence of other charged bodies. Let us assume that the charge q interacts with the system of charges q1, q2, . . . , qn. If each of the charges of the system acts on the charge q with the force F₁, F₂, ..., Fn, respectively, then the resulting force F applied to the charge q from the side of this system is equal to the vector sum of the individual forces: F = F₁ + F₂ + ... + Fn. Thus, the principle of superposition of electric fields allows us to come to one important statement.

Electric field lines

The electric field is depicted using lines of force.

Field lines indicate the direction of the force acting on a positive charge at a given point in the field.

Properties of electric field lines

Electric field lines have a beginning and an end. They start on positive charges and end on negative ones.

The lines of force of the electric field are always perpendicular to the surface of the conductor.

The distribution of electric field lines determines the nature of the field. The field can be radial(if the lines of force come out of one point or converge at one point), homogeneous(if the lines of force are parallel) and heterogeneous(if the lines of force are not parallel).

charge density- this is the amount of charge per unit length, area or volume, thus determining the linear, surface and volume charge densities, which are measured in the SI system: in Coulombs per meter (C/m), in Coulombs per square meter (C/m² ) and Coulomb per cubic meter (C/m³), respectively. Unlike the density of matter, the charge density can have both positive and negative values, this is due to the fact that there are positive and negative charges.

Linear, surface and bulk charge densities are usually denoted by the functions , and, respectively, where is the radius vector. Knowing these functions, we can determine the total charge:

§5 The flow of the intensity vector

Let us define the vector flow through an arbitrary surface dS, is the normal to the surface. α is the angle between the normal and the force line of the vector. You can enter an area vector. VECTOR FLOW called the scalar value Ф E equal to the scalar product of the intensity vector by the area vector

For a uniform field

For an inhomogeneous field

where is a projection, is a projection.

In the case of a curved surface S, it must be divided into elementary surfaces dS, calculate the flow through the elementary surface, and the total flow will be equal to the sum or, in the limit, the integral of the elementary flows

where is the integral over a closed surface S (for example, over a sphere, cylinder, cube, etc.)

The flux of a vector is an algebraic quantity: it depends not only on the configuration of the field, but also on the choice of direction. For closed surfaces, the outer normal is taken as the positive direction of the normal, i.e. a normal pointing outward of the area covered by the surface.

For a uniform field, the flux through a closed surface is zero. In the case of an inhomogeneous field

3. The intensity of the electrostatic field created by a uniformly charged spherical surface.

Let a spherical surface of radius R (Fig. 13.7) bear a uniformly distributed charge q, i.e. the surface charge density at any point on the sphere will be the same.

We enclose our spherical surface in a symmetric surface S with radius r>R. The intensity vector flux through the surface S will be equal to

According to the Gauss theorem

Hence

Comparing this relation with the formula for the field strength of a point charge, we can conclude that the field strength outside the charged sphere is as if the entire charge of the sphere were concentrated in its center.

2. Electrostatic field of the ball.

Let we have a ball of radius R, uniformly charged with bulk density.

At any point A, lying outside the ball at a distance r from its center (r> R), its field is similar to the field of a point chargelocated at the center of the ball. Then outside the ball

and on its surface (r=R)

The Ostrogradsky–Gauss theorem, which we will prove and discuss later, establishes a connection between electric charges and an electric field. It is a more general and more elegant formulation of Coulomb's law.

In principle, the strength of the electrostatic field created by a given charge distribution can always be calculated using Coulomb's law. The total electric field at any point is the vector sum (integral) contribution of all charges, i.e.

However, except for the simplest cases, it is extremely difficult to calculate this sum or integral.

Here the Ostrogradsky-Gauss theorem comes to the rescue, with the help of which it is much easier to calculate the electric field strength created by a given charge distribution.

The main value of the Ostrogradsky-Gauss theorem is that it allows deeper understanding of the nature of the electrostatic field and establishes more general relation between charge and field.

But before moving on to the Ostrogradsky-Gauss theorem, it is necessary to introduce the concepts: lines of force electrostatic field and tension vector flow electrostatic field.

In order to describe the electric field, you need to set the intensity vector at each point of the field. This can be done analytically or graphically. For this they use lines of force- these are lines, the tangent to which at any point of the field coincides with the direction of the intensity vector(Fig. 2.1).

Rice. 2.1

The line of force is assigned a certain direction - from a positive charge to a negative one, or to infinity.

Consider the case uniform electric field.

Homogeneous called an electrostatic field, at all points of which the intensity is the same in magnitude and direction, i.e. A uniform electrostatic field is depicted by parallel lines of force at an equal distance from each other (such a field exists, for example, between the plates of a capacitor) (Fig. 2.2).

In the case of a point charge, the lines of tension emanate from the positive charge and go to infinity; and from infinity enter into a negative charge. Because then the density of field lines is inversely proportional to the square of the distance from the charge. Because the surface area of ​​the sphere through which these lines pass itself increases in proportion to the square of the distance, then the total number of lines remains constant at any distance from the charge.

For a system of charges, as we see, the lines of force are directed from a positive charge to a negative one (Fig. 2.2).

Rice. 2.2

Figure 2.3 also shows that the density of field lines can serve as an indicator of the value.

The density of field lines should be such that a unit area normal to the intensity vector is crossed by such a number that is equal to the modulus of the intensity vector, i.e.