III. Electrostatics. Basic formulas. Basic formulas and guidelines for solving electrostatic problems

Where F- modulus of the force of interaction of two point charges with the value q 1 and q 2 , r- distance between charges,  - dielectric permittivity of the medium,  0 - dielectric constant.

Electric field strength

Where - force acting on a point charge q 0 placed in given point fields.

Field strength of a point charge (modulo)

Where r- distance from charge q to the point at which the tension is determined.

field strength, created by the system point charges (principle of superposition of electric fields)

Where - intensity at a given point of the field created by the i-th charge.

The modulus of the field strength created by an infinite uniformly charged plane:

Where
is the surface charge density.

Field strength modulus of a flat capacitor in its middle part

.

The formula is valid if the distance between the plates is much less than the linear dimensions of the capacitor plates.

tension field created by an infinitely long uniformly charged thread (or cylinder) at a distance r from the thread or axis of the cylinder modulo:

,

Where
- linear charge density.

a) through an arbitrary surface placed in an inhomogeneous field

,

Where  - angle between the tension vector and normal to a surface element dS- surface element area, E n- projection of the tension vector on the normal;

b) through a flat surface placed in a uniform electric field:

,

c) through a closed surface:

,

where integration is carried out over the entire surface.

Gauss theorem. The flow of the intensity vector through any closed surface S is equal to the algebraic sum of charges q 1 , q 2 ... q n covered by this surface, divided by    0 .

.

The flux of the electric displacement vector is expressed similarly to the flux of the electric field strength vector:

a) flow through a flat surface if the field is uniform

b) in the case of an inhomogeneous field and an arbitrary surface

,

Where D n- vector projection to the direction of the normal to the surface element, the area of ​​which is equal to dS.

Gauss theorem. Electric induction vector flux through a closed surface S covering the charges q 1 , q 2 ... q n, is equal to

,

Where n- the number of charges enclosed inside a closed surface (charges with their own sign).

Potential energy of a system of two point charges Q And q provided that W = 0, is found by the formula:

W=
,

Where r- distance between charges. Potential energy is positive in the interaction of like charges and negative in the interaction of unlike charges.

The potential of the electric field created by a point charge Q on distance r

 =
,

The potential of the electric field created by a metal sphere of radius R, carrying a charge Q:

 =
(r ≤ R; field inside and on the surface of the sphere),

 =
(r > R; field outside the sphere).

The potential of the electric field created by the system n point charges in accordance with the principle of superposition of electric fields is equal to the algebraic sum of potentials  1 ,  2 ,…,  n, created by charges q 1 , q 2 , ..., q n at a given point in the field

 = .

Relationship of potentials with tension:

a) in general = -qrad or =
;

b) in the case of a homogeneous field

E =
,

Where d- distance between equipotential surfaces with potentials  1 And  2 along the power line;

c) in the case of a field with central or axial symmetry

where is the derivative taken along the line of force.

The work done by the field forces to move the charge q from point 1 to point 2

A=q( 1 -  2 ),

Where (  1 -  2 ) is the potential difference between the initial and final points of the field.

The potential difference and the electric field strength are related by the relations

( 1 -  2 ) =
,

Where E e- projection of tension vector to the direction of travel dl.

The electric capacitance of a solitary conductor is determined by the charge ratio q on conductor to conductor potential  .

.

Capacitor capacitance:

,

Where (  1 -  2 ) = U- potential difference (voltage) between the capacitor plates; q- charge module on one plate of the capacitor.

Electrical capacitance of a conducting ball (sphere) in SI

c = 4 0 R,

Where R- ball radius,  - relative permittivity of the medium;  0 = 8.8510 -12 F/m.

Electric capacitance of a flat capacitor in the SI system:

,

Where S- area of ​​one plate; d- distance between plates.

Capacitance of a spherical capacitor (two concentric spheres with radii R 1 And R 2 , the space between which is filled with a dielectric, with a permittivity  ):

.

Capacitance of a cylindrical capacitor (two coaxial cylinders with a length l and radii R 1 And R 2 , the space between them is filled with a dielectric with a permittivity  )

.

Battery capacity of n capacitors connected in series is determined by the relation

.

The last two formulas are applicable to determine the capacitance of multilayer capacitors. The arrangement of layers parallel to the plates corresponds to the series connection of single-layer capacitors; if the boundaries of the layers are perpendicular to the plates, then it is considered that there is a parallel connection of single-layer capacitors.

Potential energy of a system of fixed point charges

.

Here  i- the potential of the field created at the point where the charge is located q i, by all charges except i th; n is the total number of charges.

Volumetric energy density of the electric field (energy per unit volume):

 =
= = ,

Where D- magnitude of the electric displacement vector.

Uniform field energy:

W= V.

Energy of inhomogeneous field:

W=
.

Electric charge is a physical quantity that characterizes the ability of particles or bodies to enter into electromagnetic interactions. Electric charge is usually denoted by the letters q or Q. In the SI system, electric charge is measured in Coulomb (C). A free charge of 1 C is a gigantic amount of charge, practically not found in nature. As a rule, you will have to deal with microcoulombs (1 μC = 10 -6 C), nanocoulombs (1 nC = 10 -9 C) and picocoulombs (1 pC = 10 -12 C). Electric charge has the following properties:

1. Electric charge is a kind of matter.

2. The electric charge does not depend on the movement of the particle and on its speed.

3. Charges can be transferred (for example, by direct contact) from one body to another. Unlike body mass, electric charge is not an inherent characteristic of a given body. The same body in different conditions may have different charges.

4. There are two types of electric charges, conventionally named positive And negative.

5. All charges interact with each other. At the same time, like charges repel each other, unlike charges attract. The forces of interaction of charges are central, that is, they lie on a straight line connecting the centers of charges.

6. There is the smallest possible (modulo) electric charge, called elementary charge. Its meaning:

e= 1.602177 10 -19 C ≈ 1.6 10 -19 C

The electric charge of any body is always a multiple of the elementary charge:

Where: N is an integer. Please note that it is impossible to have a charge equal to 0.5 e; 1,7e; 22,7e and so on. Physical quantities that can take only a discrete (not continuous) series of values ​​are called quantized. The elementary charge e is a quantum (the smallest portion) of the electric charge.

In an isolated system, the algebraic sum of the charges of all bodies remains constant:

The law of conservation of electric charge states that in a closed system of bodies processes of the birth or disappearance of charges of only one sign cannot be observed. It also follows from the law of conservation of charge if two bodies of the same size and shape that have charges q 1 and q 2 (it doesn’t matter what sign the charges are), bring into contact, and then back apart, then the charge of each of the bodies will become equal:

From the modern point of view, charge carriers are elementary particles. All ordinary bodies are made up of atoms, which include positively charged protons, negatively charged electrons and neutral particles neutrons. Protons and neutrons are part of atomic nuclei, the electrons form electron shell atoms. The electric charges of the proton and electron modulo are exactly the same and equal to the elementary (that is, the minimum possible) charge e.

In a neutral atom, the number of protons in the nucleus is equal to the number of electrons in the shell. This number is called the atomic number. An atom of a given substance can lose one or more electrons, or acquire an extra electron. In these cases, the neutral atom turns into a positively or negatively charged ion. Please note that positive protons are part of the nucleus of an atom, so their number can only change during nuclear reactions. Obviously, when electrifying bodies nuclear reactions not happening. Therefore, in any electrical phenomena, the number of protons does not change, only the number of electrons changes. So, giving a body a negative charge means transferring extra electrons to it. And the message of a positive charge, contrary to common mistake, does not mean the addition of protons, but the subtraction of electrons. Charge can be transferred from one body to another only in portions containing an integer number of electrons.

Sometimes in problems the electric charge is distributed over some body. To describe this distribution, the following quantities are introduced:

1. Linear charge density. Used to describe the distribution of charge along the filament:

Where: L- thread length. Measured in C/m.

2. Surface charge density. Used to describe the distribution of charge over the surface of a body:

Where: S is the surface area of ​​the body. Measured in C / m 2.

3. Bulk charge density. Used to describe the distribution of charge over the volume of a body:

Where: V- volume of the body. Measured in C / m 3.

Please note that electron mass is equal to:

me\u003d 9.11 ∙ 10 -31 kg.

Coulomb's law

point charge called a charged body, the dimensions of which can be neglected under the conditions of this problem. Based on numerous experiments, Coulomb established the following law:

The forces of interaction of fixed point charges are directly proportional to the product of charge modules and inversely proportional to the square of the distance between them:

Where: ε – dielectric permittivity of the medium – a dimensionless physical quantity showing how many times the force of electrostatic interaction in a given medium will be less than in vacuum (that is, how many times the medium weakens the interaction). Here k- coefficient in the Coulomb law, the value that determines the numerical value of the force of interaction of charges. In the SI system, its value is taken equal to:

k= 9∙10 9 m/F.

The forces of interaction of point stationary charges obey Newton's third law, and are forces of repulsion from each other with the same signs of charges and forces of attraction to each other with different signs. The interaction of fixed electric charges is called electrostatic or Coulomb interaction. The section of electrodynamics that studies the Coulomb interaction is called electrostatics.

Coulomb's law is valid for point charged bodies, uniformly charged spheres and balls. In this case, for distances r take the distance between the centers of spheres or balls. In practice, Coulomb's law is well fulfilled if the dimensions of the charged bodies are much smaller than the distance between them. Coefficient k in the SI system is sometimes written as:

Where: ε 0 \u003d 8.85 10 -12 F / m - electrical constant.

Experience shows that the forces of the Coulomb interaction obey the principle of superposition: if a charged body interacts simultaneously with several charged bodies, then the resulting force acting on this body is equal to the vector sum of the forces acting on this body from all other charged bodies.

Remember also two important definitions:

conductors- substances containing free carriers of electric charge. Inside the conductor, free movement of electrons - charge carriers is possible ( electricity). Conductors include metals, electrolyte solutions and melts, ionized gases, and plasma.

Dielectrics (insulators)- substances in which there are no free charge carriers. The free movement of electrons inside dielectrics is impossible (electric current cannot flow through them). It is dielectrics that have a certain permittivity not equal to unity ε .

For the permittivity of a substance, the following is true (about what an electric field is a little lower):

Electric field and its intensity

According to modern concepts, electric charges do not act directly on each other. Each charged body creates in the surrounding space electric field. This field has a force effect on other charged bodies. The main property of an electric field is the action on electric charges with a certain force. Thus, the interaction of charged bodies is carried out not by their direct influence on each other, but through the electric fields surrounding the charged bodies.

The electric field surrounding a charged body can be investigated using the so-called test charge - a small point charge that does not introduce a noticeable redistribution of the investigated charges. To quantify the electric field, a force characteristic is introduced - electric field strength E.

The strength of the electric field is called physical quantity, equal to the ratio of the force with which the field acts on a test charge placed at a given point of the field, to the value of this charge:

The electric field strength is a vector physical quantity. The direction of the tension vector coincides at each point in space with the direction of the force acting on the positive test charge. The electric field of stationary and unchanging charges with time is called electrostatic.

For a visual representation of the electric field, use lines of force. These lines are drawn so that the direction of the tension vector at each point coincides with the direction of the tangent to the line of force. Force lines have the following properties.

• The lines of force of an electrostatic field never intersect.
• The lines of force of an electrostatic field are always directed from positive charges to negative ones.
• When depicting an electric field using lines of force, their density should be proportional to the modulus of the field strength vector.
• The lines of force start at a positive charge, or infinity, and end at a negative charge, or infinity. The density of the lines is the greater, the greater the tension.
• At a given point in space, only one line of force can pass, because the strength of the electric field at a given point in space is uniquely specified.

An electric field is called homogeneous if the intensity vector is the same at all points in the field. For example, a flat capacitor creates a uniform field - two plates charged with an equal and opposite charge, separated by a dielectric layer, and the distance between the plates is much smaller sizes plates.

At all points of a uniform field per charge q, introduced into a uniform field with intensity E, there is a force of the same magnitude and direction equal to F = Eq. Moreover, if the charge q positive, then the direction of the force coincides with the direction of the tension vector, and if the charge is negative, then the force and tension vectors are oppositely directed.

Positive and negative point charges are shown in the figure:

Superposition principle

If an electric field created by several charged bodies is investigated using a test charge, then the resulting force turns out to be equal to the geometric sum of the forces acting on the test charge from each charged body separately. Therefore, the strength of the electric field created by the system of charges at a given point in space is equal to the vector sum of the strengths of the electric fields created at the same point by the charges separately:

This property of the electric field means that the field obeys superposition principle. In accordance with Coulomb's law, the strength of the electrostatic field created by a point charge Q on distance r from it, is equal in modulo:

This field is called the Coulomb field. In the Coulomb field, the direction of the intensity vector depends on the sign of the charge Q: If Q> 0, then the intensity vector is directed away from the charge, if Q < 0, то вектор напряженности направлен к заряду. Величина напряжённости зависит от величины заряда, среды, в которой находится заряд, и уменьшается с увеличением расстояния.

The electric field strength that a charged plane creates near its surface:

So, if in the task it is required to determine the field strength of the system of charges, then it is necessary to act according to the following algorithm:

1. Draw a drawing.
2. Draw the field strength of each charge separately at the desired point. Remember that tension is directed towards the negative charge and away from the positive charge.
3. Calculate each of the tensions using the appropriate formula.
4. Add the stress vectors geometrically (i.e. vectorially).

Potential energy of interaction of charges

Electric charges interact with each other and with an electric field. Any interaction is described by potential energy. Potential energy of interaction of two point electric charges calculated by the formula:

Pay attention to the lack of modules in the charges. For opposite charges, the interaction energy has negative meaning. The same formula is also valid for the interaction energy of uniformly charged spheres and balls. As usual, in this case the distance r is measured between the centers of balls or spheres. If there are more than two charges, then the energy of their interaction should be considered as follows: divide the system of charges into all possible pairs, calculate the interaction energy of each pair and sum up all the energies for all pairs.

Problems on this topic are solved, as well as problems on the law of conservation of mechanical energy: first, the initial interaction energy is found, then the final one. If the task asks to find the work on the movement of charges, then it will be equal to the difference between the initial and final total energy of the interaction of charges. The interaction energy can also be converted into kinetic energy or into other types of energy. If the bodies are at a very large distance, then the energy of their interaction is assumed to be 0.

Please note: if the task requires finding the minimum or maximum distance between bodies (particles) during movement, then this condition will be satisfied at the time when the particles move in the same direction at the same speed. Therefore, the solution must begin with writing the law of conservation of momentum, from which this same speed is found. And then you should write the law of conservation of energy, taking into account the kinetic energy of the particles in the second case.

Potential. Potential difference. Voltage

An electrostatic field has an important property: the work of the forces of an electrostatic field when moving a charge from one point of the field to another does not depend on the shape of the trajectory, but is determined only by the position of the start and end points and the magnitude of the charge.

A consequence of the independence of the work from the shape of the trajectory is the following statement: the work of the forces of the electrostatic field when moving the charge along any closed trajectory is equal to zero.

The property of potentiality (independence of work from the shape of the trajectory) of an electrostatic field allows us to introduce the concept of the potential energy of a charge in an electric field. And a physical quantity equal to the ratio of the potential energy of an electric charge in an electrostatic field to the value of this charge is called potential φ electric field:

Potential φ is the energy characteristic of the electrostatic field. IN international system units (SI) the unit of potential (and hence the potential difference, i.e. voltage) is the volt [V]. Potential is a scalar quantity.

In many problems of electrostatics, when calculating potentials, it is convenient to take the point at infinity as the reference point, where the values ​​of potential energy and potential vanish. In this case, the concept of potential can be defined as follows: the potential of the field at a given point in space is equal to the work that electric forces do when a unit positive charge is removed from a given point to infinity.

Recalling the formula for the potential energy of interaction of two point charges and dividing it by the value of one of the charges in accordance with the definition of the potential, we get that potential φ point charge fields Q on distance r from it relative to a point at infinity is calculated as follows:

The potential calculated by this formula can be positive or negative, depending on the sign of the charge that created it. The same formula expresses the field potential of a uniformly charged ball (or sphere) at r ≥ R(outside the ball or sphere), where R is the radius of the ball, and the distance r measured from the center of the ball.

For a visual representation of the electric field, along with lines of force, use equipotential surfaces. A surface at all points of which the potential of the electric field has the same values ​​is called an equipotential surface or a surface of equal potential. The electric field lines are always perpendicular to the equipotential surfaces. The equipotential surfaces of the Coulomb field of a point charge are concentric spheres.

Electrical voltage it's just a potential difference, i.e. the definition of electrical voltage can be given by the formula:

In a uniform electric field, there is a relationship between field strength and voltage:

The work of the electric field can be calculated as the difference between the initial and final potential energy of the system of charges:

The work of the electric field in the general case can also be calculated using one of the formulas:

In a uniform field, when a charge moves along its lines of force, the work of the field can also be calculated using the following formula:

In these formulas:

• φ is the potential of the electric field.
• ∆φ - potential difference.
• W is the potential energy of the charge in an external electric field.
• A- the work of the electric field on the movement of the charge (charges).
• q is the charge that moves in an external electric field.
• U- voltage.
• E is the electric field strength.
• d or ∆ l is the distance over which the charge is moved along the lines of force.

In all the previous formulas, it was specifically about the work of the electrostatic field, but if the task says that “the work must be done”, or in question About work external forces”, then this work should be considered in the same way as the work of the field, but with the opposite sign.

Potential superposition principle

From the principle of superposition of field strengths created by electric charges, the principle of superposition for potentials follows (in this case, the sign of the field potential depends on the sign of the charge that created the field):

Note how much easier it is to apply the principle of superposition of potential than of tension. Potential is a scalar quantity that has no direction. Adding potentials is simply summing up numerical values.

electrical capacitance. Flat capacitor

When a charge is communicated to a conductor, there is always a certain limit, more than which it will not be possible to charge the body. To characterize the ability of a body to accumulate an electric charge, the concept is introduced electrical capacitance. The capacitance of a solitary conductor is the ratio of its charge to potential:

In the SI system, capacitance is measured in Farads [F]. 1 Farad is an extremely large capacitance. For comparison, the total capacity the globe much less than one farad. The capacitance of a conductor does not depend on its charge or on the potential of the body. Similarly, the density does not depend on either the mass or the volume of the body. Capacity depends only on the shape of the body, its dimensions and the properties of its environment.

Electrical capacity system of two conductors is called a physical quantity, defined as the ratio of the charge q one of the conductors to the potential difference Δ φ between them:

The value of the electrical capacitance of the conductors depends on the shape and size of the conductors and on the properties of the dielectric separating the conductors. There are such configurations of conductors in which the electric field is concentrated (localized) only in a certain region of space. Such systems are called capacitors, and the conductors that make up the capacitor are called facings.

The simplest capacitor is a system of two flat conductive plates arranged parallel to each other at a small distance compared to the dimensions of the plates and separated by a dielectric layer. Such a capacitor is called flat. The electric field of a flat capacitor is mainly localized between the plates.

Each of the charged plates of a flat capacitor creates an electric field near its surface, the modulus of intensity of which is expressed by the ratio already given above. Then the modulus of the final field strength inside the capacitor created by two plates is equal to:

Outside the capacitor, the electric fields of the two plates are directed in different directions, and therefore the resulting electrostatic field E= 0. can be calculated using the formula:

Thus, the capacitance of a flat capacitor is directly proportional to the area of ​​the plates (plates) and inversely proportional to the distance between them. If the space between the plates is filled with a dielectric, the capacitance of the capacitor increases by ε once. note that S in this formula there is an area of ​​​​only one plate of the capacitor. When in the problem they talk about the "plate area", they mean exactly this value. You should never multiply or divide by 2.

Once again, we present the formula for capacitor charge. By the charge of a capacitor is meant only the charge of its positive lining:

Force of attraction of the capacitor plates. The force acting on each plate is determined not full field capacitor, but by the field created by the opposite plate (the plate does not act on itself). The strength of this field is equal to half the strength of the full field, and the force of interaction of the plates:

Capacitor energy. It is also called the energy of the electric field inside the capacitor. Experience shows that a charged capacitor contains a store of energy. The energy of a charged capacitor is equal to the work of external forces that must be expended to charge the capacitor. There are three equivalent forms of writing the formula for the energy of a capacitor (they follow one from the other if you use the relation q = CU):

Pay special attention to the phrase: "The capacitor is connected to the source." This means that the voltage across the capacitor does not change. And the phrase "The capacitor was charged and disconnected from the source" means that the charge of the capacitor will not change.

Electric field energy

Electrical energy should be considered as potential energy stored in a charged capacitor. According to modern concepts, the electrical energy of a capacitor is localized in the space between the capacitor plates, that is, in an electric field. Therefore, it is called the energy of the electric field. The energy of charged bodies is concentrated in space in which there is an electric field, i.e. we can talk about the energy of the electric field. For example, in a capacitor, energy is concentrated in the space between its plates. Thus, it makes sense to introduce a new physical characteristic - the volumetric energy density of the electric field. Using the example of a flat capacitor, one can obtain the following formula for the volumetric energy density (or the energy per unit volume of the electric field):

Capacitor connections

Parallel connection of capacitors- to increase capacity. Capacitors are connected by similarly charged plates, as if increasing the area of ​​equally charged plates. The voltage on all capacitors is the same, the total charge is equal to the sum charges of each of the capacitors, and the total capacitance is also equal to the sum of the capacitances of all capacitors connected in parallel. Let's write out the formulas for the parallel connection of capacitors:

At series connection of capacitors the total capacitance of a battery of capacitors is always less than the capacitance of the smallest capacitor included in the battery. A series connection is used to increase the breakdown voltage of capacitors. We write formulas for serial connection capacitors. The total capacitance of series-connected capacitors is found from the ratio:

From the law of conservation of charge it follows that the charges on adjacent plates are equal:

The voltage is equal to the sum of the voltages across the individual capacitors.

For two capacitors connected in series, the formula above will give us following expression for total capacity:

For N identical series-connected capacitors:

Conductive sphere

The field strength inside a charged conductor is zero. Otherwise, an electric force would act on the free charges inside the conductor, which would force these charges to move inside the conductor. This movement, in turn, would lead to heating of the charged conductor, which actually does not occur.

The fact that there is no electric field inside the conductor can be understood in another way: if it were, then the charged particles would again move, and they would move in such a way as to reduce this field to zero by their own field, because. in fact, they would not want to move, because any system tends to balance. Sooner or later, all the moving charges would stop exactly in that place, so that the field inside the conductor would become equal to zero.

On the surface of the conductor, the electric field strength is maximum. The magnitude of the electric field strength of a charged ball outside it decreases with distance from the conductor and is calculated using a formula similar to the formulas for the field strength of a point charge, in which the distances are measured from the center of the ball.

Since the field strength inside the charged conductor is zero, then the potential at all points inside and on the surface of the conductor is the same (only in this case, the potential difference, and hence the tension, is zero). The potential inside the charged sphere is equal to the potential on the surface. The potential outside the ball is calculated by a formula similar to the formulas for the potential of a point charge, in which the distances are measured from the center of the ball.

Radius R:

If the sphere is surrounded by a dielectric, then:

Properties of a conductor in an electric field

1. Inside the conductor, the field strength is always zero.
2. The potential inside the conductor is the same at all points and is equal to the potential of the surface of the conductor. When in the problem they say that "the conductor is charged to the potential ... V", then they mean exactly the surface potential.
3. Outside the conductor near its surface, the field strength is always perpendicular to the surface.
4. If the conductor is given a charge, then it will be completely distributed over a very thin layer near the surface of the conductor (it is usually said that the entire charge of the conductor is distributed on its surface). This is easily explained: the fact is that by imparting a charge to the body, we transfer charge carriers of the same sign to it, i.e. like charges that repel each other. This means that they will strive to scatter from each other to the maximum distance possible, i.e. accumulate at the very edges of the conductor. As a consequence, if the conductor is removed from the core, then its electrostatic properties will not change in any way.
5. Outside the conductor, the field strength is greater, the more curved the surface of the conductor. The maximum value of tension is reached near the tips and sharp breaks of the conductor surface.

Notes on solving complex problems

1. Grounding something means a connection by a conductor of this object with the Earth. In this case, the potentials of the Earth and the existing object are equalized, and the charges necessary for this run across the conductor from the Earth to the object, or vice versa. In this case, it is necessary to take into account several factors that follow from the fact that the Earth is incommensurably larger than any object located on it:

• The total charge of the Earth is conditionally equal to zero, so its potential is also equal to zero, and it will remain zero after the object is connected to the Earth. In a word, to ground means to nullify the potential of an object.
• To nullify the potential (and hence the object's own charge, which could have been both positive and negative before), the object will either have to accept or give the Earth some (possibly even a very large) charge, and the Earth will always be able to provide such an opportunity.

2. Let us repeat once again: the distance between the repelling bodies is minimal at the moment when their velocities become equal in magnitude and directed in the same direction (the relative velocity of the charges is zero). At this moment, the potential energy of the interaction of charges is maximum. The distance between the attracting bodies is maximum, also at the moment of equality of velocities directed in one direction.

3. If the problem has a system consisting of a large number charges, it is necessary to consider and describe the forces acting on a charge that is not in the center of symmetry.

Successful, diligent and responsible implementation of these three points will allow you to show an excellent result on the CT, the maximum of what you are capable of.

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Definition 1

Electrostatics is an extensive branch of electrodynamics that studies and describes electrically charged bodies at rest in a certain system.

In practice, there are two types of electrostatic charges: positive (glass on silk) and negative (ebonite on wool). The elementary charge is the minimum charge ($e = 1.6 ∙10^( -19)$ C). The charge of any physical body is a multiple of the whole number of elementary charges: $q = Ne$.

Electrification of material bodies is the redistribution of charge between bodies. Ways of electrification: touch, friction and influence.

The law of conservation of electric positive charge - in a closed concept, the algebraic sum of the charges of all elementary particles remains stable and unchanged. $q_1 + q _2 + q _3 + …..+ q_n = const$. The test charge in this case is a point positive charge.

Coulomb's law

This law was established experimentally in 1785. According to this theory, the force of interaction of two point charges at rest in a medium is always directly proportional to the product of positive modules and inversely proportional to the square of the total distance between them.

The electric field is a unique kind of matter that interacts between stable electric charges, is formed around charges, affects only charges.

Such a process of fixed point elements is completely subject to Newton's third law, and is considered the result of repulsion of particles from each other with the same force of attraction to each other. The relationship of stable electric charges in electrostatics is called the Coulomb interaction.

Coulomb's law is quite fair and accurate for charged material bodies, uniformly charged balls and spheres. In this case, the distances are mainly taken as the parameters of the centers of spaces. In practice, this law is well and quickly fulfilled if the magnitudes of the charged bodies are much less than the distance between them.

Remark 1

Conductors and dielectrics also act in an electric field.

The former represent substances containing free carriers of an electromagnetic charge. Inside the conductor, free movement of electrons can occur. These elements include solutions, metals and various melts of electrolytes, ideal gases and plasma.

Dielectrics are substances in which there can be no free carriers of electric charge. The free movement of electrons within the dielectrics themselves is impossible, since no electric current flows through them. It is these physical particles that have a permeability that is not equal to the dielectric unit.

Field lines and electrostatics

The lines of force of the initial strength of the electric field are continuous lines, the tangent points to which in each medium through which they pass completely coincide with the axis of tension.

The main characteristics of the lines of force:

• do not intersect;
• not closed;
• stable;
• the end direction is the same as the direction of the vector;
• start at $+ q$ or at infinity, end at $– q$;
• are formed near the charges (where there is more tension);
• perpendicular to the surface of the main conductor.

Definition 2

The difference in electrical potentials or voltage (Ф or $U$) is the magnitude of the potentials at the starting and ending points of the positive charge trajectory. The less the potential changes along the path, the lower the field strength as a result.

The electric field strength is always directed in the direction of decreasing the initial potential.

Figure 2. Potential energy of a system of electric charges. Author24 - online exchange of student papers

Electric capacity characterizes the ability of any conductor to accumulate the necessary electric charge on its own surface.

This parameter does not depend on the electric charge, however, it can be affected by the geometric dimensions of the conductors, their shape, location and properties of the medium between the elements.

A capacitor is a universal electrical device that helps to quickly accumulate an electric charge to transfer it to a circuit.

Electric field and its intensity

According to modern ideas of scientists, electric stable charges do not directly affect each other. Each charged physical body in electrostatics creates in environment electric field. This process has a forceful effect on other charged substances. The main property of an electric field is to act on point charges with a certain force. Thus, the interaction of positively charged particles is carried out through the fields that surround the charged elements.

This phenomenon can be investigated by means of the so-called test charge - a small electric charge that does not introduce a significant redistribution of the studied charges. For quantitative detection of the field, a force feature is introduced - the electric field strength.

The intensity is called a physical indicator, which is equal to the ratio of the force with which the field acts on the trial charge placed at a given point in the field to the magnitude of the charge itself.

The electric field strength is a vector physical quantity. The direction of the vector in this case coincides at each material point of the surrounding space with the direction of the force acting on the positive charge. The electric field of elements that do not change with time and are stationary is considered to be electrostatic.

To understand the electric field, lines of force are used, which are drawn in such a way that the direction of the main axis of tension in each system coincides with the direction of the tangent to the point.

Potential difference in electrostatics

An electrostatic field includes one important property: the work of the forces of all moving particles when moving a point charge from one point of the field to another does not depend on the direction of the trajectory, but is determined solely by the position of the initial and final lines and the charge parameter.

The result of the independence of the work from the form of movement of charges is the following statement: the functional of the forces of the electrostatic field during the transformation of the charge along any closed trajectory is always equal to zero.

Figure 4. Potentiality of the electrostatic field. Author24 - online exchange of student papers

The potentiality property of an electrostatic field helps to introduce the concept of potential and internal energy charge. And the physical parameter equal to the ratio of the potential energy in the field to the magnitude of this charge is called the constant potential of the electric field.

In many difficult tasks electrostatics when determining potentials for a reference material point, where the magnitude of the potential energy and the potential itself vanishes, it is convenient to use an infinitely distant point. In this case, the significance of the potential is defined as follows: the potential of the electric field at any point in space is equal to the work that internal forces perform when a positive unit charge is removed from a given system to infinity.