Coulomb's law in vector form is: Electric charge. Its discreteness. Law of conservation of electric charge. Coulomb's law in vector and scalar form. Coulomb's law in this form

Law of conservation of charge

Electric charges can disappear and reappear. However, two elementary charges of opposite signs always appear or disappear. For example, an electron and a positron (positive electron) annihilate when they meet, i.e. turn into neutral gamma photons. In this case, the charges -e and +e disappear. During a process called pair production, a gamma photon, entering the field of an atomic nucleus, turns into a pair of particles - an electron and a positron, and charges arise - e and + e.

Thus, the total charge of an electrically isolated system cannot change. This statement is called law of conservation of electric charge.

Note that the law of conservation of electric charge is closely related to the relativistic invariance of charge. Indeed, if the magnitude of the charge depended on its speed, then by setting charges of one sign in motion, we would change the total charge of the isolated system.

Charged bodies interact with each other, with like charges repel and unlike charges attracting.

The exact mathematical expression of the law of this interaction was established in 1785 by the French physicist C. Coulomb. Since then, the law of interaction of stationary electric charges bears his name.

A charged body, the dimensions of which can be neglected, in comparison with the distance between interacting bodies, can be taken as a point charge. As a result of his experiments, Coulomb established that:

The force of interaction in a vacuum of two stationary point charges is directly proportional to the product of these charges and inversely proportional to the square of the distance between them. The index "" of the force shows that this is the force of interaction of charges in a vacuum.

It has been established that Coulomb's law is valid at distances from up to several kilometers.

To put an equal sign, it is necessary to introduce a certain proportionality coefficient, the value of which depends on the choice of system of units:

It has already been noted that in SI the charge is measured in Cl. In Coulomb’s law, the dimension of the left side is known - the unit of force, the dimension of the right side is known - therefore the coefficient k turns out dimensional and equal. However, in SI it is customary to write this proportionality coefficient in a slightly different form:

hence

where is the farad ( F) – unit of electrical capacitance (see clause 3.3).

The quantity is called the electrical constant. This is truly a fundamental constant that appears in many electrodynamic equations.

Thus, Coulomb's law in scalar form has the form:

Coulomb's law can be expressed in vector form:

where is the radius vector connecting the charge q 2 with charge q 1,; - force acting on the charge q 1 charge side q 2. Per charge q 2 charge side q 1 force acts (Fig. 1.1)

Experience shows that the force of interaction between two given charges does not change if any other charges are placed near them.

Publications based on materials by D. Giancoli. "Physics in two volumes" 1984 Volume 2.

There is a force between electric charges. How does it depend on the magnitude of the charges and other factors?
This question was explored in the 1780s by the French physicist Charles Coulomb (1736-1806). He used torsion balances very similar to those used by Cavendish to determine the gravitational constant.
If a charge is applied to a ball at the end of a rod suspended on a thread, the rod is slightly deflected, the thread twists, and the angle of rotation of the thread will be proportional to the force acting between the charges (torsion balance). Using this device, Coulomb determined the dependence of force on the size of charges and the distance between them.

At that time, there were no instruments to accurately determine the amount of charge, but Coulomb was able to prepare small balls with a known charge ratio. If a charged conducting ball, he reasoned, is brought into contact with exactly the same uncharged ball, then the charge present on the first ball, due to symmetry, will be distributed equally between the two balls.
This gave him the ability to receive charges of 1/2, 1/4, etc. from the original one.
Despite some difficulties associated with the induction of charges, Coulomb was able to prove that the force with which one charged body acts on another small charged body is directly proportional to the electric charge of each of them.
In other words, if the charge of any of these bodies is doubled, the force will also be doubled; if the charges of both bodies are doubled at the same time, the force will become four times greater. This is true provided that the distance between the bodies remains constant.
By changing the distance between bodies, Coulomb discovered that the force acting between them is inversely proportional to the square of the distance: if the distance, say, doubles, the force becomes four times less.

So, Coulomb concluded, the force with which one small charged body (ideally a point charge, i.e. a body like a material point that has no spatial dimensions) acts on another charged body is proportional to the product of their charges Q 1 and Q 2 and is inversely proportional to the square of the distance between them:

Here k- proportionality coefficient.
This relationship is known as Coulomb's law; its validity has been confirmed by careful experiments, much more accurate than Coulomb's original, difficult to reproduce experiments. The exponent 2 is currently established with an accuracy of 10 -16, i.e. it is equal to 2 ± 2×10 -16.

Since we are now dealing with a new quantity - electric charge, we can select a unit of measurement so that the constant k in the formula is equal to one. Indeed, such a system of units was widely used in physics until recently.

We are talking about the CGS system (centimeter-gram-second), which uses the electrostatic charge unit SGSE. By definition, two small bodies, each with a charge of 1 SGSE, located at a distance of 1 cm from each other, interact with a force of 1 dyne.

Now, however, charge is most often expressed in the SI system, where its unit is the coulomb (C).
We will give the exact definition of a coulomb in terms of electric current and magnetic field later.
In the SI system the constant k has the magnitude k= 8.988×10 9 Nm 2 / Cl 2.

The charges arising during electrification by friction of ordinary objects (combs, plastic rulers, etc.) are in the order of magnitude a microcoulomb or less (1 µC = 10 -6 C).
The charge of an electron (negative) is approximately 1.602×10 -19 C. This is the smallest known charge; it is of fundamental importance and is represented by the symbol e, it is often called the elementary charge.
e= (1.6021892 ± 0.0000046)×10 -19 C, or e≈ 1.602×10 -19 Cl.

Since a body cannot gain or lose a fraction of an electron, the total charge of the body must be an integer multiple of the elementary charge. They say that the charge is quantized (that is, it can take only discrete values). However, since the electron charge e is very small, we usually do not notice the discreteness of macroscopic charges (a charge of 1 μC corresponds to approximately 10 13 electrons) and consider the charge to be continuous.

The Coulomb formula characterizes the force with which one charge acts on another. This force is directed along the line connecting the charges. If the signs of the charges are the same, then the forces acting on the charges are directed in opposite directions. If the signs of the charges are different, then the forces acting on the charges are directed towards each other.
Note that, in accordance with Newton's third law, the force with which one charge acts on another is equal in magnitude and opposite in direction to the force with which the second charge acts on the first.
Coulomb's law can be written in vector form, similar to Newton's law of universal gravitation:

Where F 12 - vector of force acting on the charge Q 1 charge side Q 2,
- distance between charges,
- unit vector directed from Q 2 k Q 1.
It should be borne in mind that the formula is applicable only to bodies the distance between which is significantly greater than their own dimensions. Ideally, these are point charges. For bodies of finite size, it is not always clear how to calculate the distance r between them, especially since the charge distribution may be non-uniform. If both bodies are spheres with a uniform charge distribution, then r means the distance between the centers of the spheres. It is also important to understand that the formula determines the force acting on a given charge from a single charge. If the system includes several (or many) charged bodies, then the resulting force acting on a given charge will be the resultant (vector sum) of the forces acting on the part of the remaining charges. The constant k in the Coulomb Law formula is usually expressed in terms of another constant, ε 0 , the so-called electrical constant, which is related to k ratio k = 1/(4πε 0). Taking this into account, Coulomb's law can be rewritten as follows:

where with the highest accuracy today

or rounded

Writing most other equations of electromagnetic theory is simplified by using ε 0 , because the 4π the final result is often shortened. Therefore, we will generally use Coulomb's Law, assuming that:

Coulomb's law describes the force acting between two charges at rest. When charges move, additional forces are created between them, which we will discuss in subsequent chapters. Here only charges at rest are considered; This section of the study of electricity is called electrostatics.

To be continued. Briefly about the following publication:

Electric field is one of two components of the electromagnetic field, which is a vector field that exists around bodies or particles with an electric charge, or that arises when the magnetic field changes.

Comments and suggestions are accepted and welcome!

The basic law of interaction of electric charges was found experimentally by Charles Coulomb in 1785. Coulomb found that the force of interaction between two small charged metal balls is inversely proportional to the square of the distance between them and depends on the magnitude of the charges And :

,

Where -proportionality factor
.

Forces acting on charges, are central , that is, they are directed along the straight line connecting the charges.

Coulomb's law can be written down in vector form:
,

Where -charge side ,

- radius vector connecting the charge with charge ;

- module of the radius vector.

Force acting on the charge from the outside equal to
,
.

Coulomb's law in this form

fair only for the interaction of point electric charges, that is, such charged bodies whose linear dimensions can be neglected in comparison with the distance between them.

expresses the strength of interaction between stationary electric charges, that is, this is the electrostatic law.

Formulation of Coulomb's law:

The force of electrostatic interaction between two point electric charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.

Proportionality factor in Coulomb's law depends

from the properties of the environment

selection of units of measurement of quantities included in the formula.

That's why can be represented by the relation
,

Where -coefficient depending only on the choice of system of units of measurement;

- a dimensionless quantity characterizing the electrical properties of the medium is called relative dielectric constant of the medium . It does not depend on the choice of system of units of measurement and is equal to one in a vacuum.

Then Coulomb's law will take the form:
,

for vacuum
,

Then
-the relative dielectric constant of a medium shows how many times in a given medium the force of interaction between two point electric charges is And , located at a distance from each other , less than in a vacuum.

In the SI system coefficient
, And

Coulomb's law has the form:
.

This rationalized notation of the law K catch.

- electrical constant,
.

In the SGSE system
,
.

In vector form, Coulomb's law takes the form

Where -vector of force acting on the charge charge side ,

- radius vector connecting the charge with charge

r–modulus of the radius vector .

Any charged body consists of many point electric charges, therefore the electrostatic force with which one charged body acts on another is equal to the vector sum of the forces applied to all point charges of the second body by each point charge of the first body.

1.3. Electric field. Tension.

Space, in which the electric charge is located has certain physical properties.

Just in case another the charge introduced into this space is acted upon by electrostatic Coulomb forces.

If a force acts at every point in space, then a force field is said to exist in that space.

The field, along with matter, is a form of matter.

If the field is stationary, that is, does not change over time, and is created by stationary electric charges, then such a field is called electrostatic.

Electrostatics studies only electrostatic fields and interactions of stationary charges.

To characterize the electric field, the concept of intensity is introduced . Tensionyu at each point of the electric field is called the vector , numerically equal to the ratio of the force with which this field acts on a test positive charge placed at a given point and the magnitude of this charge, and directed in the direction of the force.

Test charge, which is introduced into the field, is assumed to be a point charge and is often called a test charge.

- He does not participate in the creation of the field, which is measured with its help.

It is assumed that this charge does not distort the field being studied, that is, it is small enough and does not cause a redistribution of charges that create the field.

If on a test point charge the field acts by force , then the tension
.

Tension units:

SI:

SSSE:

In the SI system expression For point charge fields:

.

In vector form:

Here – radius vector drawn from the charge q, creating a field at a given point.

T
in this way electric field strength vectors of a point chargeq at all points of the field are directed radially(Fig. 1.3)

- from the charge, if it is positive, “source”

- and to the charge if it is negative"drain"

For graphical interpretation electric field is introduced concept of a line of force orlines of tension . This

curve , the tangent at each point to which coincides with the tension vector.

The voltage line starts at a positive charge and ends at a negative charge.

The tension lines do not intersect, since at each point of the field the tension vector has only one direction.

· valid only for the interaction of point electric charges, that is, such charged bodies whose linear dimensions can be neglected in comparison with the distance between them.

· expresses the strength of interaction between stationary electric charges, that is, this is the electrostatic law.

Formulation of Coulomb's law:

The force of electrostatic interaction between two point electric charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.

Proportionality factor in Coulomb's law depends

1. from the properties of the environment

2. selection of units of measurement of quantities included in the formula.

Therefore, it can be represented by the relation,

Where - coefficient depending only on the choice of system of units of measurement;

The dimensionless quantity characterizing the electrical properties of the medium is called relative dielectric constant of the medium . It does not depend on the choice of system of units of measurement and is equal to one in a vacuum.

Then Coulomb’s law will take the form: ,

for vacuum,

Then - The relative dielectric constant of a medium shows how many times in a given medium the force of interaction between two point electric charges and located at a distance from each other is less than in a vacuum.

In the SI system coefficient , and

Coulomb's law has the form: .

This rationalized notation of the law K catch.

Electric constant, .

In the SGSE system , .

In vector form, Coulomb's law takes the form

Where - vector of force acting on the charge from the side of the charge ,

- radius vector connecting charge to charge

r–modulus of the radius vector.

Any charged body consists of many point electric charges, therefore the electrostatic force with which one charged body acts on another is equal to the vector sum of the forces applied to all point charges of the second body by each point charge of the first body.

1.3. Electric field. Tension.

Space, in which the electric charge is located has certain physical properties.

1. Just in case another the charge introduced into this space is acted upon by electrostatic Coulomb forces.

2. If a force acts at every point in space, then they say that there is a force field in this space.

3. The field, along with matter, is a form of matter.

4. If the field is stationary, that is, does not change over time, and is created by stationary electric charges, then such a field is called electrostatic.

Electric charge. Its discreteness. Law of conservation of electric charge. Coulomb's law in vector and scalar form.

Electric charge is a physical quantity that characterizes the property of particles or bodies to enter into electromagnetic force interactions. Electric charge is usually denoted by the letters q or Q. There are two kinds of electric charges, conventionally called positive and negative. Charges can be transferred (for example, by direct contact) from one body to another. Unlike body mass, electric charge is not an integral characteristic of a given body. The same body under different conditions can have a different charge. Like charges repel, unlike charges attract. The electron and proton are carriers of elementary negative and positive charges, respectively. The unit of electric charge is a coulomb (C) - an electric charge passing through the cross section of a conductor at a current of 1 A in 1 s.

Electric charge is discrete, i.e. the charge of any body is an integer multiple of the elementary electric charge e ().

Law of conservation of charge: the algebraic sum of the electric charges of any closed system (a system that does not exchange charges with external bodies) remains unchanged: q1 + q2 + q3 + ... +qn = const.

Coulomb's law: The force of interaction between two point electric charges is proportional to the magnitude of these charges and inversely proportional to the square of the distance between them.

(in scalar form)

Where F - Coulomb force, q1 and q2 - Electric charge of the body, r - Distance between charges, e0 = 8.85*10^(-12) - Electric constant, e - Dielectric constant of the medium, k = 9*10^9 - Proportionality factor.

For Coulomb’s law to be satisfied, 3 conditions are necessary:

Condition 1: Pointedness of charges - that is, the distance between charged bodies is much greater than their sizes

Condition 2: Immobility of charges. Otherwise, additional effects come into force: the magnetic field of a moving charge and the corresponding additional Lorentz force acting on another moving charge

Condition 3: Interaction of charges in a vacuum

In vector form the law is written as follows:

Where is the force with which charge 1 acts on charge 2; q1, q2 - magnitude of charges; - radius vector (vector directed from charge 1 to charge 2, and equal, in absolute value, to the distance between charges - ); k - proportionality coefficient.

Electrostatic field strength. Expression for the electrostatic field strength of a point charge in vector and scalar form. Electric field in vacuum and matter. The dielectric constant.

The electrostatic field strength is a vector force characteristic of the field and is numerically equal to the force with which the field acts on a unit test charge introduced at a given point in the field:

The unit of tension is 1 N/C - this is the intensity of an electrostatic field that acts on a charge of 1 C with a force of 1 N. The tension is also expressed in V/m.

As follows from the formula and Coulomb’s law, the field strength of a point charge in a vacuum

or

The direction of vector E coincides with the direction of the force that acts on the positive charge. If the field is created by a positive charge, then vector E is directed along the radius vector from the charge into external space (repulsion of the test positive charge); if the field is created by a negative charge, then vector E is directed towards the charge.

That. tension is a force characteristic of an electrostatic field.

For a graphical representation of the electrostatic field, vector strength lines are used ( power lines). The density of the field lines can be used to judge the magnitude of the tension.

If the field is created by a system of charges, then the resulting force acting on a test charge introduced at a given point in the field is equal to the geometric sum of the forces acting on the test charge from each point charge separately. Therefore, the intensity at a given point of the field is equal to:

This ratio expresses principle of field superposition: the strength of the resulting field created by a system of charges is equal to the geometric sum of the field strengths created at a given point by each charge separately.

Electric current in a vacuum can be created by the ordered movement of any charged particles (electrons, ions).

The dielectric constant- a quantity characterizing the dielectric properties of a medium - its response to an electric field.

In most dielectrics in not very strong fields, the dielectric constant does not depend on the field E. In strong electric fields (comparable to intra-atomic fields), and in some dielectrics in ordinary fields, the dependence of D on E is nonlinear. Also, the dielectric constant shows how many times the interaction force F between electric charges in a given medium is less than their interaction force Fo in a vacuum

The relative dielectric constant of a substance can be determined by comparing the capacitance of a test capacitor with a given dielectric (Cx) and the capacitance of the same capacitor in a vacuum (Co):

The principle of superposition as a fundamental property of fields. General expressions for the strength and potential of the field created at a point with a radius vector by a system of point charges located at points with coordinates (see paragraph 4)

If we consider the principle of superposition in the most general sense, then according to it, the sum of the influence of external forces acting on a particle will be the sum of the individual values ​​of each of them. This principle applies to various linear systems, i.e. systems whose behavior can be described by linear relationships. An example would be a simple situation where a linear wave propagates in a specific medium, in which case its properties will be preserved even under the influence of disturbances arising from the wave itself. These properties are defined as a specific sum of the effects of each of the harmonious components.

The principle of superposition can take other formulations that are completely equivalent to the above:

· The interaction between two particles does not change when a third particle is introduced, which also interacts with the first two.

· The interaction energy of all particles in a many-particle system is simply the sum of the energies of pair interactions between all possible pairs of particles. There are no many-particle interactions in the system.

· The equations describing the behavior of a many-particle system are linear in the number of particles.

6 Circulation of the voltage vector is the work done by electric forces when moving a single positive charge along a closed path L

Since the work of the electrostatic field forces along a closed loop is zero (the work of the potential field forces), therefore the circulation of the electrostatic field strength along a closed loop is zero.

Field potential. The work of any electrostatic field when moving a charged body in it from one point to another also does not depend on the shape of the trajectory, just like the work of a uniform field. On a closed trajectory, the work of the electrostatic field is always zero. Fields with this property are called potential. In particular, the electrostatic field of a point charge has a potential character.
The work of a potential field can be expressed in terms of a change in potential energy. The formula is valid for any electrostatic field.

7-11If the field lines of a uniform electric field with intensity penetrate a certain area S, then the flow of the intensity vector (previously we called the number of field lines through the area) will be determined by the formula:

where En is the product of the vector and the normal to a given area (Fig. 2.5).

Rice. 2.5

The total number of lines of force passing through the surface S is called the flux of the FU intensity vector through this surface.

In vector form, we can write the scalar product of two vectors, where vector .

Thus, the vector flux is a scalar, which, depending on the value of the angle α, can be either positive or negative.

Let's look at the examples shown in Figures 2.6 and 2.7.

	Rice. 2.6	Rice. 2.7

For Figure 2.6, surface A1 is surrounded by a positive charge and the flow here is directed outward, i.e. The surface A2– is surrounded by a negative charge, here it is directed inward. The total flux through surface A is zero.

For Figure 2.7, the flux will not be zero if the total charge inside the surface is not zero. For this configuration, the flux through surface A is negative (count the number of field lines).

Thus, the flux of the voltage vector depends on the charge. This is the meaning of the Ostrogradsky-Gauss theorem.

Gauss's theorem

The experimentally established Coulomb's law and the superposition principle make it possible to fully describe the electrostatic field of a given system of charges in a vacuum. However, the properties of the electrostatic field can be expressed in another, more general form, without resorting to the idea of ​​a Coulomb field of a point charge.

Let us introduce a new physical quantity characterizing the electric field – the flow Φ of the electric field strength vector. Let there be some fairly small area ΔS located in the space where the electric field is created. The product of the vector modulus by the area ΔS and the cosine of the angle α between the vector and the normal to the site is called the elementary flux of the intensity vector through the site ΔS (Fig. 1.3.1):

Let us now consider some arbitrary closed surface S. If we divide this surface into small areas ΔSi, determine the elementary flows ΔΦi of the field through these small areas, and then sum them up, then as a result we obtain the flow Φ of the vector through the closed surface S (Fig. 1.3.2 ):

Gauss's theorem states:

The flow of the electrostatic field strength vector through an arbitrary closed surface is equal to the algebraic sum of the charges located inside this surface, divided by the electric constant ε0.

where R is the radius of the sphere. The flux Φ through a spherical surface will be equal to the product of E and the area of ​​the sphere 4πR2. Hence,

Let us now surround the point charge with an arbitrary closed surface S and consider an auxiliary sphere of radius R0 (Fig. 1.3.3).

Consider a cone with a small solid angle ΔΩ at the apex. This cone will highlight a small area ΔS0 on the sphere, and an area ΔS on the surface S. The elementary fluxes ΔΦ0 and ΔΦ through these areas are the same. Really,

In a similar way, it can be shown that if a closed surface S does not cover a point charge q, then the flow Φ = 0. Such a case is depicted in Fig. 1.3.2. All lines of force of the electric field of a point charge penetrate the closed surface S through and through. There are no charges inside the surface S, so in this region the field lines do not break off or arise.

A generalization of Gauss's theorem to the case of an arbitrary charge distribution follows from the superposition principle. The field of any charge distribution can be represented as a vector sum of the electric fields of point charges. The flow Φ of a system of charges through an arbitrary closed surface S will be the sum of the flows Φi of the electric fields of individual charges. If the charge qi happens to be inside the surface S, then it makes a contribution to the flow equal to if this charge is outside the surface, then the contribution of its electric field to the flow will be equal to zero.

Thus, Gauss's theorem is proven.

Gauss's theorem is a consequence of Coulomb's law and the principle of superposition. But if we take the statement contained in this theorem as the original axiom, then its consequence will be Coulomb’s law. Therefore, Gauss's theorem is sometimes called an alternative formulation of Coulomb's law.

Using Gauss's theorem, in some cases it is possible to easily calculate the electric field strength around a charged body if the given charge distribution has some symmetry and the general structure of the field can be guessed in advance.

An example is the problem of calculating the field of a thin-walled, hollow, uniformly charged long cylinder of radius R. This problem has axial symmetry. For reasons of symmetry, the electric field must be directed along the radius. Therefore, to apply Gauss’s theorem, it is advisable to choose a closed surface S in the form of a coaxial cylinder of some radius r and length l, closed at both ends (Fig. 1.3.4).

For r ≥ R, the entire flux of the intensity vector will pass through the side surface of the cylinder, the area of ​​which is equal to 2πrl, since the flux through both bases is zero. Application of Gauss's theorem gives:

This result does not depend on the radius R of the charged cylinder, so it also applies to the field of a long uniformly charged filament.

To determine the field strength inside a charged cylinder, it is necessary to construct a closed surface for the case r< R. В силу симметрии задачи поток вектора напряженности через боковую поверхность гауссова цилиндра должен быть и в этом случае равен Φ = E 2πrl. Согласно теореме Гаусса, этот поток пропорционален заряду, оказавшемуся внутри замкнутой поверхности. Этот заряд равен нулю. Отсюда следует, что электрическое поле внутри однородно заряженного длинного полого цилиндра равно нулю.

In a similar way, one can apply Gauss's theorem to determine the electric field in a number of other cases when the distribution of charges has some kind of symmetry, for example, symmetry about the center, plane or axis. In each of these cases, it is necessary to choose a closed Gaussian surface of an appropriate shape. For example, in the case of central symmetry, it is convenient to choose a Gaussian surface in the form of a sphere with the center at the symmetry point. With axial symmetry, the closed surface must be chosen in the form of a coaxial cylinder, closed at both ends (as in the example discussed above). If the distribution of charges does not have any symmetry and the general structure of the electric field cannot be guessed, the application of Gauss's theorem cannot simplify the problem of determining the field strength.

Let's consider another example of a symmetrical charge distribution - determining the field of a uniformly charged plane (Fig. 1.3.5).

In this case, it is advisable to choose the Gaussian surface S in the form of a cylinder of some length, closed at both ends. The axis of the cylinder is directed perpendicular to the charged plane, and its ends are located at the same distance from it. Due to symmetry, the field of a uniformly charged plane must be directed along the normal everywhere. Application of Gauss's theorem gives:

where σ is the surface charge density, i.e. charge per unit area.

The resulting expression for the electric field of a uniformly charged plane is also applicable in the case of flat charged areas of finite size. In this case, the distance from the point at which the field strength is determined to the charged area should be significantly less than the size of the area.

And schedules for 7 – 11

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

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

a. Let us enclose our spherical surface in a symmetrical surface S with radius r>R. The flux of the tension vector through the surface S will be equal to

By Gauss's theorem

Hence

c. Let us draw through point B, located inside a charged spherical surface, a sphere S of radius r

2. Electrostatic field of the ball.

Let us have a ball of radius R, uniformly charged with volume 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 charge located in the center of the ball. Then out of the ball

(13.10)

and on its surface (r=R)

(13.11)

At point B, lying inside the ball at a distance r from its center (r>R), the field is determined only by the charge enclosed inside the sphere with radius r. The flux of the tension vector through this sphere is equal to

on the other hand, in accordance with Gauss's theorem

From a comparison of the last expressions it follows

(13.12)

where is the dielectric constant inside the ball. The dependence of the field strength created by a charged sphere on the distance to the center of the ball is shown in (Fig. 13.10)

Let us assume that a hollow cylindrical surface of radius R is charged with a constant linear density.

Let us draw a coaxial cylindrical surface of radius. The flow of the tension vector through this surface

By Gauss's theorem

From the last two expressions we determine the field strength created by a uniformly charged thread:

(13.13)

Let the plane have infinite extent and the charge per unit area equal to σ. From the laws of symmetry it follows that the field is directed everywhere perpendicular to the plane, and if there are no other external charges, then the fields on both sides of the plane must be the same. Let us limit part of the charged plane to an imaginary cylindrical box, so that the box is cut in half and its constituents are perpendicular, and the two bases, each having an area S, are parallel to the charged plane (Figure 1.10).

Total vector flow; tension is equal to the vector multiplied by the area S of the first base, plus the flux of the vector through the opposite base. The tension flux through the side surface of the cylinder is zero, because lines of tension do not intersect them. Thus, On the other hand, according to Gauss's theorem

Hence

but then the field strength of an infinite uniformly charged plane will be equal to

(13.14)

This expression does not include coordinates, therefore the electrostatic field will be uniform, and its intensity at any point in the field will be the same.

5. The field strength created by two infinite parallel planes charged oppositely with the same densities.

As can be seen from Figure 13.13, the field strength between two infinite parallel planes having surface charge densities and is equal to the sum of the field strengths created by the plates, i.e.

Thus,

(13.15)

Outside the plate, the vectors from each of them are directed in opposite directions and cancel each other out. Therefore, the field strength in the space surrounding the plates will be zero E=0.

12. Field of a uniformly charged sphere.

Let the electric field be created by a charge Q, uniformly distributed over the surface of a sphere of radius R(Fig. 190). To calculate the field potential at an arbitrary point located at a distance r from the center of the sphere, it is necessary to calculate the work done by the field when moving a unit positive charge from a given point to infinity. Previously, we proved that the field strength of a uniformly charged sphere outside it is equivalent to the field of a point charge located in the center of the sphere. Consequently, outside the sphere, the field potential of the sphere will coincide with the field potential of a point charge

φ (r)=Q 4πε 0r . (1)

In particular, on the surface of the sphere the potential is equal to φ 0=Q 4πε 0R. There is no electrostatic field inside the sphere, so the work done to move a charge from an arbitrary point located inside the sphere to its surface is zero A= 0, therefore the potential difference between these points is also zero Δ φ = -A= 0. Consequently, all points inside the sphere have the same potential, coinciding with the potential of its surface φ 0=Q 4πε 0R .

So, the distribution of the field potential of a uniformly charged sphere has the form (Fig. 191)

φ (r)=⎧⎩⎨Q 4πε 0R, npu r<RQ 4πε 0r, npu r>R . (2)

Please note that there is no field inside the sphere, and the potential is non-zero! This example is a clear illustration of the fact that the potential is determined by the value of the field from a given point to infinity.

Dipole.

A dielectric (like any substance) consists of atoms and molecules. Since the positive charge of all the nuclei of the molecule is equal to the total charge of the electrons, the molecule as a whole is electrically neutral.

The first group of dielectrics(N 2, H 2, O 2, CO 2, CH 4, ...) are substances whose molecules have a symmetrical structure, i.e., the centers of “gravity” of positive and negative charges in the absence of an external electric field coincide and, therefore, the dipole moment of the molecule R equal to zero.Molecules such dielectrics are called non-polar. Under the influence of an external electric field, the charges of non-polar molecules are shifted in opposite directions (positive along the field, negative against the field) and the molecule acquires a dipole moment.

For example, a hydrogen atom. In the absence of a field, the center of the negative charge distribution coincides with the position of the positive charge. When the field is turned on, the positive charge shifts in the direction of the field, the negative charge moves against the field (Fig. 6):

Figure 6

Model of a non-polar dielectric - elastic dipole (Fig. 7):

Figure 7

The dipole moment of this dipole is proportional to the electric field

The second group of dielectrics(H 2 O, NH 3, SO 2, CO,...) are substances whose molecules have asymmetrical structure, i.e. the centers of “gravity” of positive and negative charges do not coincide. Thus, these molecules have a dipole moment in the absence of an external electric field. Molecules such dielectrics are called polar. In the absence of an external field, however, The dipole moments of polar molecules due to thermal motion are randomly oriented in space and their resulting moment is zero. If such a dielectric is placed in an external field, then the forces of this field will tend to rotate the dipoles along the field and a nonzero resulting torque arises.

Polar - the centers of “+” charge and the centers of “-” charge are displaced, for example, in the water molecule H 2 O.

Model of a polar dielectric rigid dipole:

Figure 8

Dipole moment of the molecule:

The third group of dielectrics(NaCl, KCl, KBr, ...) are substances whose molecules have an ionic structure. Ionic crystals are spatial lattices with regular alternation of ions of different signs. In these crystals it is impossible to isolate individual molecules, but they can be considered as a system of two ionic sublattices pushed into one another. When an electric field is applied to an ionic crystal, some deformation of the crystal lattice or a relative displacement of the sublattices occurs, leading to the appearance of dipole moments.

Product of charge | Q| dipole on his shoulder l called electric dipole moment:

p=|Q|l.

Dipole field strength

Where R- electric dipole moment; r- module of the radius vector drawn from the center of the dipole to the point at which the field strength interests us; α- angle between radius vector r and shoulder l dipoles (Fig. 16.1).

The dipole field strength at a point lying on the dipole axis (α=0),

and at a point lying perpendicular to the dipole arm, raised from its middle () .

Dipole field potential

The dipole field potential at a point lying on the dipole axis (α = 0),

and at a point lying perpendicular to the dipole arm, raised from its middle () , φ = 0.

Mechanical moment, acting on a dipole with an electric moment R, placed in a uniform electric field with intensity E,

M=[p;E](vector multiplication), or M=pE sinα ,

where α is the angle between the directions of the vectors R And E.

· current strength I (serves as a quantitative measure of electric current) - a scalar physical quantity determined by the electric charge passing through the cross section of a conductor per unit time:

· current density - physical quantity determined by the strength of the current passing through a unit cross-sectional area of ​​a conductor perpendicular to the direction of the current

- vector, oriented in the direction of the current (i.e. the direction of the vector j coincides with the direction of the ordered movement of positive charges.

The unit of current density is ampere per meter squared (A/m2).

Current strength through an arbitrary surface S is defined as the flow of the vector j, i.e.

· Expression for current density in terms of the average speed of current carriers and their concentration

During the time dt, charges will pass through the platform dS, spaced from it no further than vdt (the expression for the distance between the charges and the platform in terms of speed)

Charge dq passed through dS during dt

where q 0 is the charge of one carrier; n is the number of charges per unit volume (i.e.

concentration): dS·v·dt - volume.

hence, the expression for the current density in terms of the average speed of current carriers and their concentration has the following form:

· D.C.– a current whose strength and direction do not change over time.

Where q- electric charge passing over time t through the cross section of the conductor. The unit of current is ampere (A).

· external forces and EMF of the current source

outside forces - strength non-electrostatic origin, acting on charges from current sources.

External forces do work to move electric charges.

These forces are electromagnetic in nature:

and their work on transferring test charge q is proportional to q:

· A physical quantity determined by the work done by external forces when moving a unit positive charge is calledelectromotive force (emf), acting in the circuit:

where e is called the electromotive force of the current source. The “+” sign corresponds to the case when, when moving, the source passes in the direction of the action of external forces (from the negative plate to the positive), “-” - to the opposite case

· Ohm's law for a circuit section