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

Electric field strength

Where - force acting on a point charge q 0 , placed at a given point in the field.

Field strength of a point charge (modulo)

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

Field strength created by a system of point charges (principle of superposition of electric fields)

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

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

Where
- 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 cylinder axis modulo:

,

Where
- linear charge density.

a) through an arbitrary surface placed in a non-uniform field

,

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

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

,

c) through a closed surface:

,

where integration is carried out over the entire surface.

Gauss's theorem. Flow of a tension vector through any closed surface S 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 a non-uniform field and an arbitrary surface

,

Where D n- vector projection to the direction of the normal to a surface element whose area is equal to dS.

Gauss's theorem. Electrical induction vector flow through a closed surface S, covering charges q 1 , q 2 ... q n, is equal

,

Where n- the number of charges contained 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, found by the formula:

W=
,

Where r- distance between charges. Potential energy is positive when like charges interact and negative when unlike charges interact.

Electric field potential created by a point charge Q at a distance r

 =
,

Electric field potential created metal sphere radius R, carrying charge Q:

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

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

Electric field potential 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 between potentials and tension:

a) in general = -qrad or =
;

b) in the case of a uniform 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 a central or axial symmetry

where is the derivative is taken along the force line.

Work done by field forces to move a charge q from point 1 to point 2

A = q( 1 -  2 ),

Where (  1 -  2 ) - potential difference between the starting and ending points of the field.

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

( 1 -  2 ) =
,

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

The electrical capacity of an isolated conductor is determined by the charge ratio q on the conductor to the conductor potential  .

.

Capacitance of the capacitor:

,

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

Electrical capacity of a conducting ball (sphere) in SI

c = 4 0 R,

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

Electrical capacity of a flat capacitor in the SI system:

,

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

Electrical capacity 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 dielectric constant  ):

.

Electrical capacity of a cylindrical capacitor (two coaxial cylinders length l and radii R 1 And R 2 , the space between which is filled with a dielectric with dielectric constant  )

.

Battery capacity from 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 the 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 stationary point charges

.

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

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

 =
= = ,

Where D- the magnitude of the electrical displacement vector.

Uniform field energy:

W= V.

Non-uniform field energy:

W=
.

Encyclopedic YouTube

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  The foundation of electrostatics was laid by the work of Coulomb (although ten years before him, the same results, even with even greater accuracy, were obtained by Cavendish. The results of Cavendish’s work were kept in the family archive and were published only a hundred years later); The law of electrical interactions discovered by the latter made it possible for Green, Gauss and Poisson to create a mathematically elegant theory. The most essential part of electrostatics is the theory of potential, created by Green and Gauss. A lot of experimental research on electrostatics was carried out by Rees, whose books in the past constituted the main guide for the study of these phenomena.

  Permittivity

  Finding the value of the dielectric coefficient K of any substance, a coefficient included in almost all formulas that one has to deal with in electrostatics, can be done quite in various ways. The most commonly used methods are the following.

  1) Comparison of the electrical capacities of two capacitors having same sizes and shape, but in one of them the insulating layer is a layer of air, in the other - a layer of the dielectric being tested.

  2) Comparison of attractions between the surfaces of a capacitor, when a certain potential difference is imparted to these surfaces, but in one case there is air between them (attractive force = F 0), in the other case, the test liquid insulator (attractive force = F). The dielectric coefficient is found by the formula:

  K = F 0 F . (\displaystyle K=(\frac (F_(0))(F)).)

  3) Observations of electric waves (see Electrical oscillations) propagating along wires. According to Maxwell's theory, the speed of propagation of electric waves along wires is expressed by the formula

  V = 1 K μ . (\displaystyle V=(\frac (1)(\sqrt (K\mu ))).)

  in which K denotes the dielectric coefficient of the medium surrounding the wire, μ denotes the magnetic permeability of this medium. We can put μ = 1 for the vast majority of bodies, and therefore it turns out

  V = 1 K. (\displaystyle V=(\frac (1)(\sqrt (K))).)

  Usually, the lengths of standing electric waves that arise in parts of the same wire located in the air and in the test dielectric (liquid) are compared. Having determined these lengths λ 0 and λ, we obtain K = λ 0 2 / λ 2. According to Maxwell’s theory, it follows that when an electric field is excited in any insulating substance, special deformations occur inside this substance. Along the induction tubes, the insulating medium is polarized. Electrical displacements arise in it, which can be likened to the movements of positive electricity along the axes of these tubes, and through each cross section the amount of electricity passing through the tube is equal to

  D = 1 4 π K F . (\displaystyle D=(\frac (1)(4\pi ))KF.)

  Maxwell's theory makes it possible to find expressions for those internal forces (forces of tension and pressure) that appear in dielectrics when an electric field is excited in them. This question was first considered by Maxwell himself, and later in more detail by Helmholtz. Further development of the theory of this issue and the closely connected theory of electrostriction (that is, the theory that considers phenomena that depend on the occurrence of special voltages in dielectrics when an electric field is excited in them) belongs to the works of Lorberg, Kirchhoff, P. Duhem, N. N. Schiller and some others

  Boundary conditions

  Let's finish summary The most significant part of the department of electrostriction is the consideration of the question of refraction of induction tubes. Let us imagine two dielectrics in an electric field, separated from each other by some surface S, with dielectric coefficients K 1 and K 2.

  Let at points P 1 and P 2 located infinitely close to the surface S on either side of it, the magnitudes of the potentials are expressed through V 1 and V 2 , and the magnitudes of the forces experienced by a unit of positive electricity placed at these points through F 1 and F 2. Then for a point P lying on the surface S itself, there must be V 1 = V 2,

  d V 1 d s = d V 2 d s , (30) (\displaystyle (\frac (dV_(1))(ds))=(\frac (dV_(2))(ds)),\qquad (30))

  if ds represents an infinitesimal displacement along the line of intersection of the tangent plane to the surface S at point P with the plane passing through the normal to the surface at this point and through the direction of the electric force in it. On the other hand, it should be

  K 1 d V 1 d n 1 + K 2 d V 2 d n 2 = 0. (31) (\displaystyle K_(1)(\frac (dV_(1))(dn_(1)))+K_(2)( \frac (dV_(2))(dn_(2)))=0.\qquad (31))

  Let us denote by ε 2 the angle made by the force F2 with the normal n2 (inside the second dielectric), and by ε 1 the angle made by the force F 1 with the same normal n 2 Then, using formulas (31) and (30), we find

  t g ε 1 t g ε 2 = K 1 K 2 . (\displaystyle (\frac (\mathrm (tg) (\varepsilon _(1)))(\mathrm (tg) (\varepsilon _(2))))=(\frac (K_(1))(K_( 2))).)

  So, on the surface separating two dielectrics from each other, the electric force undergoes a change in its direction, like a light ray entering from one medium into another. This consequence of the theory is justified by experience.

  ... All predictions of electrostatics follow from its two laws.
  But it is one thing to express these things mathematically, and quite another to
  apply them with ease and with just the right amount of wit.
  Richard Feynman

  Electrostatics studies the interaction of stationary charges. Key experiments in electrostatics were carried out in the 17th and 18th centuries. Happy opening electromagnetic phenomena and the revolution in technology that they produced, interest in electrostatics was lost for some time. However, modern scientific research shows the enormous importance of electrostatics for understanding many processes of living and inanimate nature.

  Electrostatics and life

  In 1953, American scientists S. Miller and G. Urey showed that one of the “building blocks of life” - amino acids - can be obtained by passing an electric discharge through a gas similar in composition to the primitive atmosphere of the Earth, consisting of methane, ammonia, hydrogen and vapor water. Over the next 50 years, other researchers repeated these experiments and obtained the same results. When short current pulses are passed through bacteria, pores appear in their shell (membrane), through which DNA fragments of other bacteria can pass in, triggering one of the mechanisms of evolution. Thus, the energy required for the origin of life on Earth and its evolution could indeed be the electrostatic energy of lightning discharges (Fig. 1).

  How electrostatics cause lightning

  At any given time, about 2,000 lightning flashes at different points on the Earth, approximately 50 lightning strikes the Earth every second, and every square kilometer of the Earth's surface is struck by lightning on average six times a year. Back in the 18th century, Benjamin Franklin proved that lightning striking from thunderclouds is electrical discharges that carry negative charge. Moreover, each of the discharges supplies the Earth with several tens of coulombs of electricity, and the amplitude of the current during a lightning strike ranges from 20 to 100 kiloamperes. High-speed photography showed that a lightning strike lasts only tenths of a second and that each lightning consists of several shorter ones.

  Using measuring instruments installed on atmospheric probes, at the beginning of the 20th century, the Earth's electric field was measured, the intensity of which at the surface turned out to be approximately 100 V/m, which corresponds to a total charge of the planet of about 400,000 C. The carrier of charges in the Earth's atmosphere are ions, the concentration of which increases with altitude and reaches a maximum at an altitude of 50 km, where under the influence cosmic radiation An electrically conductive layer - the ionosphere - was formed. Therefore, we can say that the Earth's electric field is the field of a spherical capacitor with an applied voltage of about 400 kV. Under the influence of this voltage from upper layers a current of 2–4 kA flows into the lower ones all the time, the density of which is (1–2) 10 –12 A/m 2, and energy is released up to 1.5 GW. And if there were no lightning, this electric field would disappear! It turns out that in good weather The Earth's electrical capacitor is discharged, and during a thunderstorm it is charged.

  A thundercloud is huge amount steam, some of which has condensed in the form of tiny droplets or pieces of ice. The top of a thundercloud can be at an altitude of 6–7 km, and the bottom can hang above the ground at an altitude of 0.5–1 km. Above 3–4 km, the clouds consist of ice floes different sizes, since the temperature there is always below zero. These pieces of ice are in constant motion caused by rising currents warm air rising from below from the heated surface of the earth. Small pieces of ice are lighter than large ones, and they are carried away by rising air currents and collide with large ones along the way. With each such collision, electrification occurs, in which large pieces of ice are charged negatively, and small ones - positively. Over time, positively charged small pieces of ice collect mainly in the upper part of the cloud, and negatively charged large ones - at the bottom (Fig. 2). In other words, the top of the cloud is charged positively, and the bottom - negatively. In this case, positive charges are induced on the ground directly below the thundercloud. Now everything is ready for a lightning discharge, in which air breakdown occurs and the negative charge from the bottom of the thundercloud flows to the Earth.

  

  It is typical that before a thunderstorm, the strength of the Earth's electric field can reach 100 kV/m, i.e., 1000 times higher than its value in good weather. As a result, the positive charge of each hair on the head of a person standing under a thundercloud increases by the same amount, and they, pushing away from each other, stand on end (Fig. 3).

  

  Fulgurite - trace of lightning on the ground

  During a lightning discharge, energy of the order of 10 9 –10 10 J is released. Most of this energy is spent on thunder, heating the air, flash of light and the radiation of other electromagnetic waves, and only a small part is released in the place where the lightning enters the ground. But even this “small” part is enough to cause a fire, kill a person or destroy a building. Lightning can heat the channel through which it moves to 30,000°C, which is much higher than the melting point of sand (1600–2000°C). Therefore, lightning, striking the sand, melts it, and the hot air and water vapor, expanding, form a tube from the molten sand, which after some time hardens. This is how fulgurites (thunder arrows, devil's fingers) are born - hollow cylinders made of melted sand (Fig. 4). The longest excavated fulgurites went underground to a depth of more than five meters.

  

  How electrostatics protect against lightning

  Fortunately, most lightning strikes occur between clouds and therefore do not pose a threat to human health. However, it is believed that lightning kills more than a thousand people around the world every year. At least in the USA, where such statistics are kept, about a thousand people suffer from lightning strikes every year and more than a hundred of them die. Scientists have long tried to protect people from this “punishment of God.” For example, the inventor of the first electric capacitor (Leyden jar), Pieter van Muschenbrouck, in an article on electricity written for the famous French Encyclopedia, defended traditional ways preventing lightning - bell ringing and firing cannons, which he believed were quite effective.

  In 1750, Franklin invented the lightning rod. In an attempt to protect the Maryland capitol building from a lightning strike, he attached a thick iron rod to the building, extending several meters above the dome and connected to the ground. The scientist refused to patent his invention, wanting it to begin serving people as soon as possible. The mechanism of action of a lightning rod is easy to explain if we remember that the electric field strength near the surface of a charged conductor increases with increasing curvature of this surface. Therefore, under a thundercloud near the tip of the lightning rod, the field strength will be so high that it will cause ionization of the surrounding air and corona discharge in it. As a result, the probability of lightning hitting the lightning rod will increase significantly. Thus, knowledge of electrostatics not only made it possible to explain the origin of lightning, but also to find a way to protect against them.

  The news of Franklin's lightning rod quickly spread throughout Europe, and he was elected to all academies, including the Russian one. However, in some countries the devout population greeted this invention with indignation. The very idea that a person could so easily and simply tame the main weapon of God’s wrath seemed blasphemous. Therefore in different places people, for pious reasons, broke lightning rods.

  A curious incident occurred in 1780 in a small town in northern France, where the townspeople demanded that the iron lightning rod mast be demolished and the matter came to trial. The young lawyer, who defended the lightning rod from the attacks of obscurantists, based his defense on the fact that both the human mind and his ability to conquer the forces of nature are of divine origin. Everything that helps save a life is for the good, the young lawyer argued. He won the case and gained great fame. The lawyer's name was... Maximilian Robespierre.

  Well, now the portrait of the inventor of the lightning rod is the most coveted reproduction in the world, because it adorns the well-known hundred dollar bill.

  Electrostatics that brings life back

  The energy from the capacitor discharge not only led to the emergence of life on Earth, but can also restore life to people whose heart cells have stopped beating synchronously. Asynchronous (chaotic) contraction of heart cells is called fibrillation. Fibrillation of the heart can be stopped by passing a short pulse of current through all its cells. To do this, two electrodes are applied to the patient’s chest, through which a pulse is passed with a duration of about ten milliseconds and an amplitude of up to several tens of amperes. In this case, the discharge energy through chest can reach 400 J (which is equal to the potential energy of a pound weight raised to a height of 2.5 m). Device providing electrical discharge A device that stops cardiac fibrillation is called a defibrillator. The simplest defibrillator is an oscillating circuit consisting of a capacitor with a capacity of 20 μF and a coil with an inductance of 0.4 H. By charging the capacitor to a voltage of 1–6 kV and discharging it through the coil and the patient, whose resistance is about 50 ohms, you can obtain the current pulse necessary to bring the patient back to life.

  Electrostatics giving light

  A fluorescent lamp can serve as a convenient indicator of electric field strength. To verify this, while in a dark room, rub the lamp with a towel or scarf - as a result, the outer surface of the lamp glass will be charged positively, and the fabric - negatively. As soon as this happens, we will see flashes of light appearing in those places of the lamp that we touch with a charged cloth. Measurements have shown that the electric field strength inside a working fluorescent lamp is about 10 V/m. At this intensity, free electrons have the necessary energy to ionize mercury atoms inside a fluorescent lamp.

  

  Electric field under high voltage lines power transmission lines - power lines - can reach very high values. Therefore, if in the dark fluorescent lamp stick it into the ground under the power line, it will light up, and quite brightly (Fig. 5). So, using the energy of an electrostatic field, you can illuminate the space under power lines.

  How electrostatics warn of fire and make smoke cleaner

  In most cases, when choosing the type of fire alarm detector, preference is given to a smoke sensor, since a fire is usually accompanied by the release of large quantity smoke and this type of detector is able to warn people in the building about the danger. Smoke detectors use ionization or photoelectric principle to detect smoke in the air.

  Ionization smoke detectors contain an α-radiation source (usually americium-241) that ionizes the air between metal electrode plates, electrical resistance between which is constantly measured using a special circuit. The ions formed as a result of α-radiation provide conductivity between the electrodes, and the microparticles of smoke that appear there bind to the ions, neutralize their charge and thus increase the resistance between the electrodes, which reacts electrical diagram, sounding an alarm. Sensors based on this principle demonstrate very impressive sensitivity, reacting even before the very first sign of smoke is detected by a living creature. It should be noted that the radiation source used in the sensor does not pose any danger to humans, since alpha rays cannot pass even through a sheet of paper and are completely absorbed by a layer of air several centimeters thick.

  The ability of dust particles to electrify is widely used in industrial electrostatic dust collectors. A gas containing, for example, soot particles, rising upward, passes through a negatively charged metal mesh, as a result of which these particles acquire a negative charge. Continuing to rise upward, the particles find themselves in the electric field of positively charged plates, to which they are attracted, after which the particles fall into special containers, from where they are periodically removed.

  Bioelectrostatics

  One of the causes of asthma is the waste products of dust mites (Fig. 6) - insects about 0.5 mm in size that live in our house. Research has shown that asthma attacks are caused by one of the proteins that these insects secrete. The structure of this protein resembles a horseshoe, both ends of which are positively charged. The electrostatic repulsive forces between the ends of such a horseshoe-shaped protein make its structure stable. However, the properties of a protein can be changed by neutralizing its positive charges. This can be done by increasing the concentration of negative ions in the air using any ionizer, for example a Chizhevsky chandelier (Fig. 7). At the same time, the frequency of asthma attacks decreases.

  

  Electrostatics helps not only to neutralize proteins secreted by insects, but also to catch them themselves. It has already been said that the hair “stands on end” if it is charged. You can imagine what insects experience when they find themselves electrically charged. The finest hairs on their paws diverge into different sides, and insects lose the ability to move. The cockroach trap shown in Figure 8 is based on this principle. Cockroaches are attracted to sweet powder that is previously electrostatically charged. Powder (it is white in the picture) is used to cover the inclined surface around the trap. Once on the powder, the insects become charged and roll into the trap.

  

  What are antistatic agents?

  Clothes, carpets, bedspreads, etc. objects are charged after contact with other objects, and sometimes simply with jets of air. In everyday life and at work, charges generated in this way are often called static electricity.

  

  Under normal atmospheric conditions, natural fibers (cotton, wool, silk and viscose) absorb moisture well (hydrophilic) and therefore slightly conduct electricity. When such fibers touch or rub against other materials, excess electrical charges appear on their surfaces, but for a very short time, as the charges immediately flow back through the wet fibers of the fabric containing various ions.

  Unlike natural fibers, synthetic fibers (polyester, acrylic, polypropylene) do not absorb moisture well (hydrophobic), and there are fewer mobile ions on their surfaces. Upon contact synthetic materials with each other they are charged with opposite charges, but since these charges drain very slowly, the materials stick to each other, creating inconvenience and unpleasant sensations. By the way, the hair structure is very close to synthetic fibers and are also hydrophobic, so upon contact, for example, with a comb, they become charged with electricity and begin to repel each other.

  To get rid of static electricity, the surface of clothing or other items can be lubricated with a substance that retains moisture and thereby increases the concentration of mobile ions on the surface. After such treatment, the resulting electrical charge will quickly disappear from the surface of the object or be distributed over it. The hydrophilicity of a surface can be increased by lubricating it with surfactants, whose molecules are similar to soap molecules - one part of a very long molecule is charged, and the other is not. Substances that prevent the appearance of static electricity are called antistatic agents. For example, ordinary coal dust or soot is an antistatic agent, therefore, in order to get rid of static electricity, so-called lamp black is included in the impregnation of carpeting and upholstery materials. For the same purposes, up to 3% natural fibers and sometimes thin metal threads are added to such materials.

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

  1. Electric charge is a type of matter.

  2. The electric charge does not depend on the movement of the particle and 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 integral characteristic of a given body. The same body different conditions may have a different charge.

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

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

  6. There is a minimum 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– an integer. Please note that it is impossible for a charge equal to 0.5 to exist e; 1,7e; 22,7e and so on. Physical quantities that can only take a discrete (not continuous) series of values ​​are called quantized. Elementary charge e is a quantum (smallest portion) 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 creation or disappearance of charges of only one sign cannot be observed. It also follows from the law of conservation of charge that if two bodies of the same size and shape having charges q 1 and q 2 (it doesn’t matter at all what sign the charges are), bring into contact, and then separate again, then the charge of each of the bodies will become equal:

  From a modern point of view, charge carriers are elementary particles. All ordinary bodies consist of atoms, which include positively charged protons, negatively charged electrons and neutral particles - neutrons. Protons and neutrons are part of atomic nuclei, electrons form the electron shell of atoms. The electric charges of a proton and an electron are exactly the same in absolute value 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 may lose one or more electrons, or gain 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. It is obvious that when bodies are electrified, nuclear reactions do not occur. Therefore, in any electrical phenomena, the number of protons does not change, only the number of electrons changes. Thus, imparting a negative charge to a body means transferring extra electrons to it. And the message of a positive charge, contrary to common mistake, does not mean adding protons, but subtracting 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 a certain 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– body surface area. Measured in C/m2.

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

  Where: V– body volume. Measured in C/m3.

  Please note that electron mass is equal to:

  m e= 9.11∙10 –31 kg.

  Coulomb's law

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

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

  

  Where: ε – dielectric constant of a medium is a dimensionless physical quantity that shows how many times the force of electrostatic interaction in a given medium will be less than in a vacuum (that is, how many times the medium weakens the interaction). Here k– coefficient in Coulomb’s law, a 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 between point fixed charges obey Newton's third law, and are forces of repulsion from each other when the charges have the same signs and forces of attraction to each other when different signs. The interaction of stationary electric charges is called electrostatic or Coulomb interaction. The branch 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 satisfied if the sizes of charged bodies are much smaller than the distance between them. Coefficient k in the SI system it is sometimes written as:

  

  Where: ε 0 = 8.85∙10 –12 F/m – electrical constant.

  Experience shows that the Coulomb interaction forces obey the principle of superposition: if a charged body interacts simultaneously with several charged bodies, then the resulting force acting on given 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 electric charge carriers. Inside the conductor, the free movement of electrons - charge carriers - can flow through the conductors. electric current). Conductors include metals, solutions and melts of electrolytes, 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 dielectric constant, not equal to unity. ε .

  For the dielectric constant of a substance, the following is true (about what an electric field is just below):

  

  Electric field and its intensity

  According to modern concepts, electric charges do not act on each other directly. Each charged body creates in the surrounding space electric field. This field exerts a force on other charged bodies. The main property of the electric field is the effect on electric charges with some 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 studied using a so-called test charge - a small point charge that does not introduce a noticeable redistribution of the charges being studied. To quantitatively determine the electric field, a force characteristic is introduced - electric field strength E.

  Electric field strength is a physical quantity equal to the ratio of the force with which the field acts on a test charge placed at a given point in the field to the magnitude of this charge:

  

  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 charges that do not change over time is called electrostatic.

  For visual representation electric field is used power lines. 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. Field lines have the following properties.

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

  An electric field is called uniform if the intensity vector is the same at all points of the field. For example, a uniform field is created by a flat capacitor - two plates charged with a charge of equal magnitude and opposite sign, separated by a dielectric layer, and the distance between the plates is large smaller sizes plates

  At all points of a uniform field on a charge q, introduced into a uniform field with intensity E, a force of equal magnitude and direction acts, 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 studied 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. Consequently, the electric field strength created by a system of charges at a given point in space is equal to the vector sum of the electric field strengths created at the same point by 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 at a distance r from it, is equal in modulus:

  

  This field is called Coulomb field. In a Coulomb field, the direction of the intensity vector depends on the sign of the charge Q: If Q> 0, then the voltage 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 created by a charged plane near its surface:

  So, if the problem requires determining the field strength of a system of charges, then we must proceed as follows algorithm:

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

  Potential energy of charge interaction

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

  Please note that the charges have no modules. For unlike charges, the interaction energy has negative value. The same formula is 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 the balls or spheres. If there are not two, but more charges, then the energy of their interaction should be calculated 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 conservation law mechanical energy: first the initial energy of interaction is found, then the final one. If the problem asks you to find the work done by moving charges, then it will be equal to the difference between the initial and final total energy of interaction of charges. Interaction energy can also be converted into kinetic energy or other types of energy. If the bodies are very long distance, then the energy of their interaction is assumed to be equal to 0.

  Please note: if the problem requires finding the minimum or maximum distance between bodies (particles) when moving, then this condition will be met at that moment in time when the particles move in one direction at the same speed. Therefore, the solution must begin by writing down the law of conservation of momentum, from which this identical speed is found. And then we should write the law of conservation of energy, taking into account the kinetic energy of particles in the second case.

  Potential. Potential difference. Voltage

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

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

  The property of potentiality (independence of work from the shape of the trajectory) of the electrostatic field allows us to introduce the concept of 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 magnitude of this charge is called potential φ electric field:

  Potential φ is the energy characteristic of the electrostatic field. In the International System of Units (SI), the unit of potential (and therefore 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 field potential at a given point in space is equal to the work done by electric forces when removing a single positive charge 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 potential, we obtain that potential φ point charge fields Q at a distance r from it relative to the point at infinity is calculated as follows:

  The potential calculated using 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.

  To visually represent the electric field, along with lines of force, use equipotential surfaces. A surface at all points of which the electric field potential has same values, is called an equipotential surface or surface of equal potential. Electric field lines are always perpendicular to 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. definition electrical voltage can be given by the formula:

  

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

  Electric field work can be calculated as the difference between the initial and final potential energy of a 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 field lines, the work of the field can also be calculated using the following formula:

  In these formulas:

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

  In all previous formulas we were talking specifically about the work of the electrostatic field, but if the problem says that “work must be done,” or we're talking about about the “work of 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):

  Notice 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 adding up numerical values.

  Electrical capacity. Flat capacitor

  When imparting a charge to a conductor, there is always a certain limit beyond which it will not be possible to charge the body. To characterize the ability of a body to accumulate electric charge, the concept is introduced electrical capacitance. The capacitance of an isolated 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 capacitance of the entire globe is significantly less than one farad. The capacitance of a conductor depends neither on its charge nor on the potential of the body. Similarly, density does not depend on either the mass or volume of the body. Capacity depends only on the shape of the body, its size and the properties of its environment.

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

  

  The magnitude of the electrical capacitance of conductors depends on the shape and size of the conductors and on the properties of the dielectric separating the conductors. There are 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 linings.

  The simplest capacitor is a system of two flat conducting plates located parallel to each other at a small distance compared to the size of the plates and separated by a dielectric layer. Such a capacitor is called flat. The electric field of a parallel-plate 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 which is expressed by the relationship already given above. Then the modulus of the final field strength inside the capacitor created by the 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 electrical capacity 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 the area of ​​​​only one capacitor plate. When they talk about the “plating area” in a problem, they mean exactly this value. You never need to multiply or divide it by 2.

  Once again we present the formula for capacitor charge. The charge of a capacitor is understood only as the charge on its positive plate:

  The force of attraction between the capacitor plates. The force acting on each plate is determined not by the total field of the 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 total field, and the force of interaction between the plates is:

  Capacitor energy. It is also called the energy of the electric field inside the capacitor. Experience shows that a charged capacitor contains a reserve 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 we use the relation q = C.U.):

  

  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 ideas, electrical energy of the capacitor is localized in the space between the plates of the capacitor, that is, in the electric field. Therefore it is called electric field energy. 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, a capacitor's energy is concentrated in the space between its plates. Thus, it makes sense to introduce a new physical characteristics– volumetric energy density of the electric field. Using a flat capacitor as an example, we 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. The capacitors are connected by similarly charged plates, as if increasing the area of ​​the equally charged plates. The voltage on all capacitors is the same, the total charge is equal to the sum of the 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 down the formulas for parallel connection capacitors:

  

  At series connection of capacitors the total capacity of a capacitor bank is always less than the capacity of the smallest capacitor included in the battery. A series connection is used to increase the breakdown voltage of the capacitors. Let's write down the formulas for serial connection capacitors. The total capacitance of series-connected capacitors is found from the relationship:

  

  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 on the individual capacitors.

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

  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 happen.

  The fact that there is no electric field inside the conductor can be understood in another way: if there was one, then the charged particles would again move, and they would move exactly in such a way as to reduce this field to zero with their own field, because in fact, they would not want to move, because every system strives for balance. Sooner or later, all moving charges would stop exactly in that place so that the field inside the conductor would become 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 its boundaries decreases with distance from the conductor and is calculated using a formula similar to the formula for the field strength of a point charge, in which distances are measured from the center of the ball.

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

  Radius R:

  If the ball is surrounded by a dielectric, then:

  Properties of a conductor in an electric field

  1. Inside a 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 they say in a problem that “the conductor is charged to a potential ... V,” they mean precisely the surface potential.
  3. Outside the conductor near its surface, the field strength is always perpendicular to the surface.
  4. If a charge is given to a conductor, then it will all be distributed over a very thin layer near the surface of the conductor (usually they say that the entire charge of the conductor is distributed on its surface). This is easily explained: the fact is that when imparting a charge to a body, we transfer to it charge carriers of the same sign, i.e. like charges that repel each other. This means they will try to run away from each other to the maximum possible distance, i.e. accumulate at the very edges of the conductor. As a result, if the core is removed from a conductor, its electrostatic properties will not change in any way.
  5. Outside the conductor, the more curved the surface of the conductor, the greater the field strength. The maximum value of tension is achieved near the edges and sharp breaks in the surface of the conductor.

  Notes on solving complex problems

  1. Grounding something means the connection of 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 move along 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 disproportionately larger than any object located on it:

  • The total charge of the Earth is conventionally zero, so its potential is also zero, and it will remain zero after the object connects with the Earth. In a word, to ground means to reset the potential of an object.
  • To reset the potential (and therefore the object’s own charge, which could previously have been either positive or negative), the object will have to either accept or give to the Earth some (perhaps even a very large) charge, and the Earth will always be able to provide this possibility.

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

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

• Learn all the formulas and laws in physics, and formulas and methods in mathematics. In fact, this is also very simple to do; there are only about 200 necessary formulas in physics, and even a little less in mathematics. Each of these items contains about a dozen standard methods problem solving basic level difficulties that can also be learned, and thus, completely automatically and without difficulty, solve most of the CT at the right time. After this, you will only have to think about the most difficult tasks.
• Attend all three stages of rehearsal testing in physics and mathematics. Each RT can be visited twice to decide on both options. Again, on the CT, in addition to the ability to quickly and efficiently solve problems, and knowledge of formulas and methods, you must also be able to properly plan time, distribute forces, and most importantly, fill out the answer form correctly, without confusing the numbers of answers and problems, or your own last name. Also, during RT, it is important to get used to the style of asking questions in problems, which may seem very unusual to an unprepared person at the DT.

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

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In electrostatics, one of the fundamental ones is Coulomb's law. It is used in physics to determine the force of interaction between two stationary point charges or the distance between them. This is a fundamental law of nature that does not depend on any other laws. Then the shape of the real body does not affect the magnitude of the forces. In this article we will tell in simple language Coulomb's law and its application in practice.

History of discovery

Sh.O. Coulomb in 1785 was the first to experimentally prove the interactions described by the law. In his experiments he used special torsion balances. However, back in 1773, Cavendish proved, using the example of a spherical capacitor, that there is no electric field inside the sphere. This indicated that electrostatic forces vary depending on the distance between the bodies. To be more precise - the square of the distance. His research was not published then. Historically, this discovery was named after Coulomb, and the quantity in which charge is measured has a similar name.

Formulation

The definition of Coulomb's law states: In a vacuumF interaction of two charged bodies is directly proportional to the product of their moduli and inversely proportional to the square of the distance between them.

It sounds short, but may not be clear to everyone. In simple words: The more charge the bodies have and the closer they are to each other, the greater the force.

And vice versa: If you increase the distance between the charges, the force will become less.

The formula for Coulomb's rule looks like this:

Designation of letters: q - charge value, r - distance between them, k - coefficient, depends on the chosen system of units.

The charge value q can be conditionally positive or conditionally negative. This division is very arbitrary. When bodies come into contact, it can be transmitted from one to another. It follows from this that the same body can have a charge of different magnitude and sign. A point charge is a charge or body whose dimensions are much smaller than the distance of possible interaction.

It is worth considering that the environment in which the charges are located affects the F interaction. Since it is almost equal in air and vacuum, Coulomb’s discovery is applicable only for these media; this is one of the conditions for the use of this type of formula. As already mentioned, in the SI system the unit of measurement of charge is Coulomb, abbreviated Cl. It characterizes the amount of electricity per unit time. It is derived from the SI base units.

1 C = 1 A*1 s

It is worth noting that the dimension of 1 C is redundant. Due to the fact that carriers repel each other, it is difficult to contain them in a small body, although the 1A current itself is small if it flows in a conductor. For example, in the same 100 W incandescent lamp a current of 0.5 A flows, and in an electric heater it flows more than 10 A. Such a force (1 C) is approximately equal to the 1 ton mass acting on a body from the side of the globe.

You may have noticed that the formula is almost the same as in gravitational interaction, only if masses appear in Newtonian mechanics, then charges appear in electrostatics.

Coulomb formula for a dielectric medium

The coefficient, taking into account the values ​​of the SI system, is determined in N 2 * m 2 / Cl 2. It is equal to:



In many textbooks, this coefficient can be found in the form of a fraction:



Here E 0 = 8.85*10-12 C2/N*m2 is the electrical constant. For a dielectric, E is added - the dielectric constant of the medium, then Coulomb's law can be used to calculate the forces of interaction of charges for vacuum and medium.

Taking into account the influence of the dielectric, it has the form:



From this we see that the introduction of a dielectric between the bodies reduces the force F.

How are the forces directed?

Charges interact with each other depending on their polarity - like charges repel, and unlike (opposite) charges attract.





By the way, this is the main difference from a similar law of gravitational interaction, where bodies always attract. The forces are directed along the line drawn between them, called the radius vector. In physics it is denoted as r 12 and as the radius vector from the first to the second charge and vice versa. The forces are directed from the center of the charge to the opposite charge along this line if the charges are opposite, and in the opposite direction if they are of the same name (two positive or two negative). In vector form:



The force applied to the first charge by the second is denoted as F 12. Then, in vector form, Coulomb’s law looks like this:



To determine the force applied to the second charge, the designations F 21 and R 21 are used.

If the body has complex shape and it is large enough that at a given distance it cannot be considered a point charge, then it is divided into small sections and each section is considered a point charge. After geometrically adding all the resulting vectors, the resulting force is obtained. Atoms and molecules interact with each other according to the same law.

Application in practice

Coulomb's work is very important in electrostatics; in practice, it is used in a number of inventions and devices. A striking example You can select a lightning rod. With its help, they protect buildings and electrical installations from thunderstorms, thereby preventing fire and equipment failure. When it rains with a thunderstorm, an induced charge of large magnitude appears on the ground, they are attracted towards the cloud. It turns out that a large electric field appears on the surface of the earth. Near the tip of the lightning rod it is larger, as a result of which a corona discharge is ignited from the tip (from the ground, through the lightning rod to the cloud). The charge from the ground is attracted to the opposite charge of the cloud, according to Coulomb's law. The air is ionized, and the electric field strength decreases near the end of the lightning rod. Thus, charges do not accumulate on the building, in which case the likelihood of a lightning strike is low. If a strike does occur on the building, then all the energy will go into the ground through the lightning rod.

Serious scientific research uses the greatest device of the 21st century - the particle accelerator. In it, the electric field does work to increase the energy of the particle. Considering these processes from the point of view of the influence of a group of charges on a point charge, then all the relations of the law turn out to be valid.

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