ELECTROSTATIC FIELD

electrostatic field test charge q 0

tension

, (4)

, . (5)

power lines

WORK OF ELECTROSTATIC FIELD FORCES. POTENTIAL

The electric field, like the gravitational field, is potential. Those. the work performed by electrostatic forces does not depend on the route along which the charge q is moved in the electric field from point 1 to point 2. This work is equal to the difference in the potential energies possessed by the moving charge at the initial and final points of the field:

A 1,2 = W 1 – W 2. (7)

It can be shown that the potential energy of a charge q is directly proportional to the magnitude of this charge. Therefore, as an energy characteristic of an electrostatic field, the ratio of the potential energy of a test charge q 0 placed at any point in the field to the value of this charge is used:

This quantity represents the amount of potential energy per unit of positive charge and is called field potential V given point. [φ] = J / Cl = V (Volts).

If we accept that when charge q 0 moves away to infinity (r→ ∞), its potential energy in the field of charge q becomes zero, then the potential of the field of a point charge q at a distance r from it:

. (9)

If a field is created by a system of point charges, then the potential of the resulting field is equal to the algebraic (including signs) sum of the potentials of each of them:

. (10)

From the definition of potential (8) and expression (7), the work done by the forces of the electrostatic field to move the charge from

point 1 to point 2 can be represented as:

ELECTRIC CURRENT IN GASES

SELF-SELF-SUSTAINED GAS DISCHARGE

Gases are good insulators at temperatures that are not too high and at pressures close to atmospheric. If placed in dry atmospheric air, a charged electrometer, its charge remains unchanged for a long time. This is explained by the fact that gases under normal conditions consist of neutral atoms and molecules and do not contain free charges (electrons and ions). A gas becomes a conductor of electricity only when some of its molecules are ionized. To ionize, the gas must be exposed to some kind of ionizer: for example, an electric discharge, x-ray radiation, radiation or UV radiation, candle flame, etc. (in the latter case, the electrical conductivity of the gas is caused by heating).

During the ionization of gases, one or more electrons are removed from the outer electron shell of an atom or molecule, which leads to the formation of free electrons and positive ions. Electrons can attach to neutral molecules and atoms, turning them into negative ions. Therefore, an ionized gas contains positively and negatively charged ions and free electrons. E Electric current in gases is called gas discharge. Thus, the current in gases is created by ions of both signs and electrons. A gas discharge with such a mechanism will be accompanied by the transfer of matter, i.e. Ionized gases are classified as conductors of the second type.

In order to remove one electron from a molecule or atom, it is necessary to perform a certain amount of work A and, i.e. expend some energy. This energy is called ionization energy , the values ​​of which for atoms of various substances lie in the range of 4÷25 eV. The ionization process is usually characterized quantitatively by a quantity called ionization potential :

Simultaneously with the process of ionization in a gas, the reverse process always occurs - the process of recombination: positive and negative ions or positive ions and electrons, meeting, reunite with each other to form neutral atoms and molecules. The more ions appear under the influence of the ionizer, the more intense the recombination process.

Strictly speaking, the electrical conductivity of a gas is never zero, since it always contains free charges formed as a result of the action of radiation from radioactive substances present on the surface of the Earth, as well as cosmic radiation. The intensity of ionization under the influence of these factors is low. This insignificant electrical conductivity of the air causes the leakage of charges from electrified bodies, even if they are well insulated.

The nature of the gas discharge is determined by the composition of the gas, its temperature and pressure, the size, configuration and material of the electrodes, as well as the applied voltage and current density.

Let us consider a circuit containing a gas gap (Fig.), subjected to continuous, constant-intensity exposure to an ionizer. As a result of the action of the ionizer, the gas acquires some electrical conductivity and current flows in the circuit. Figure shows the current-voltage characteristics (current versus applied voltage) for two ionizers. Performance
(the number of ion pairs produced by the ionizer in the gas gap in 1 second) of the second ionizer is greater than the first. We will assume that the productivity of the ionizer is constant and equal to n 0. At not very low pressure, almost all of the detached electrons are captured by neutral molecules, forming negatively charged ions. Taking into account recombination, we assume that the concentrations of ions of both signs are the same and equal to n. The average drift velocities of ions of different signs in an electric field are different: , . b - and b + – mobility of gas ions. Now for region I, taking into account (5), we can write:

As can be seen, in region I, with increasing voltage, the current increases, as the drift speed increases. The number of pairs of recombining ions will decrease with increasing speed.

Region II - the region of saturation current - all ions created by the ionizer reach the electrodes without having time to recombine. Saturation current density

j n = q n 0 d, (28)

where d is the width of the gas gap (the distance between the electrodes). As can be seen from (28), the saturation current is a measure of the ionizing effect of the ionizer.

At a voltage greater than U p p (region III), the speed of electrons reaches such a value that when they collide with neutral molecules they are capable of causing impact ionization. As a result, additional An 0 ion pairs are formed. The quantity A is called the gas gain coefficient . In region III, this coefficient does not depend on n 0, but depends on U. Thus. the charge reaching the electrodes at constant U is directly proportional to the performance of the ionizer - n 0 and the voltage U. For this reason, region III is called the proportionality region. U pr – proportionality threshold. The gas amplification factor A has values ​​from 1 to 10 4.

In region IV, the region of partial proportionality, the gas gain coefficient begins to depend on n 0. This dependence increases with increasing U. The current increases sharply.

In the voltage range 0 ÷ U g, current in the gas exists only when the ionizer is active. If the action of the ionizer is stopped, the discharge also stops. Discharges that exist only under the influence of external ionizers are called non-self-sustaining.

Voltage Ug is the threshold of the region, the Geiger region, which corresponds to the state when the process in the gas gap does not disappear even after the ionizer is turned off, i.e. the discharge acquires the character of an independent discharge. Primary ions only give impetus to the occurrence of a gas discharge. In this region, massive ions of both signs also acquire the ability to ionize. The magnitude of the current does not depend on n 0 .

In region VI, the voltage is so high that the discharge, once occurring, does not stop - the region of continuous discharge.

SELF-INDEPENDENT GAS DISCHARGE AND ITS TYPES

A discharge in a gas that persists after the external ionizer stops working is called self-discharge.

Let us consider the conditions for the occurrence of a self-sustained discharge. At high voltages (regions V–VI), electrons generated under the influence of an external ionizer, strongly accelerated by the electric field, collide with neutral gas molecules and ionize them. As a result, secondary electrons and positive ions are formed (process 1 in Fig. 158). Positive ions move towards the cathode and electrons move towards the anode. The secondary electrons re-ionize the gas molecules, and therefore the total number of electrons and ions will increase as the electrons move toward the anode in an avalanche fashion. This causes an increase in electric current (see Fig. Area V). The described process is called impact ionization.

However, impact ionization under the influence of electrons is not enough to maintain the discharge when the external ionizer is removed. To do this, it is necessary that electron avalanches be “reproduced,” that is, that new electrons arise in the gas under the influence of some processes. Such processes are shown schematically in Fig. 158: Positive ions accelerated by the field, hitting the cathode, knock electrons out of it (process 2); Positive ions, colliding with gas molecules, transfer them to an excited state, the transition of such molecules to a normal state is accompanied by the emission of a photon (process 3); A photon absorbed by a neutral molecule ionizes it, the so-called process of photon ionization of molecules occurs (process 4); Knockout of electrons from the cathode under the influence of photons (process 5).

Finally, at significant voltages between the electrodes of the gas gap, a moment comes when positive ions, which have a shorter free path than electrons, acquire energy sufficient to ionize gas molecules (process 6), and ion avalanches rush to the negative plate. When, in addition to electron avalanches, ion avalanches also occur, the current strength increases practically without an increase in voltage (region VI in the figure).

As a result of the described processes, the number of ions and electrons in the gas volume increases like an avalanche, and the discharge becomes independent, i.e., it persists even after the termination of the external ionizer. The voltage at which a self-discharge occurs is called the breakdown voltage. For air, this is about 30,000 V for every centimeter of distance.

Depending on the gas pressure, the configuration of the electrodes, and the parameters of the external circuit, we can talk about four types of independent discharge: glow, spark, arc and corona.

1. Glow discharge. Occurs at low pressures. If you apply to the electrodes soldered into a glass tube 30÷50 cm long constant pressure at several hundred volts, gradually pumping air out of the tube, then at a pressure of ≈ 5.3÷6.7 kPa, a discharge occurs in the form of a luminous, winding reddish cord running from the cathode to the anode. With a further decrease in pressure, the cord thickens, and at a pressure of ≈ 13 Pa the discharge has the form schematically shown in Fig.

Directly adjacent to the cathode is a thin luminous layer 1 - the first cathode glow, or cathode film, followed by a dark layer 2 - the cathode dark space, which then passes into the luminous layer 3 - a smoldering glow, which has a sharp boundary on the cathode side, gradually disappearing on the anode side. It occurs due to the recombination of electrons with positive ions. The smoldering glow is bordered by a dark gap 4 - the Faraday dark space, followed by a column of ionized luminous gas 5 - the positive column. The positive column does not have a significant role in maintaining the discharge. For example, when the distance between the electrodes of the tube decreases, its length decreases, while the cathode parts of the discharge remain unchanged in shape and size. In a glow discharge, only two parts of it are of particular importance for its maintenance: the cathode dark space and the glow. In the cathode dark space, there is a strong acceleration of electrons and positive ions, knocking electrons out of the cathode (secondary emission). In the region of smoldering glow, impact ionization of gas molecules by electrons occurs. The positive ions formed in this case rush to the cathode and knock out new electrons from it, which, in turn, again ionize the gas, etc. In this way, the glow discharge is continuously maintained.

With further pumping of the tube at a pressure of ≈ 1.3 Pa, the glow of the gas weakens and the walls of the tube begin to glow. Electrons knocked out of the cathode by positive ions at such rarefaction rarely collide with gas molecules and therefore, accelerated by the field, hitting the glass, causing it to glow, the so-called cathodoluminescence. The flow of these electrons was historically called cathode rays.

Glow discharge is widely used in technology. Since the glow of the positive column has a color characteristic of each gas, it is used in gas-light tubes for luminous inscriptions and advertisements (for example, neon gas-discharge tubes give a red glow, argon - bluish-green). In fluorescent lamps, which are more economical than incandescent lamps, the glow discharge radiation occurring in mercury vapor is absorbed by a fluorescent substance (phosphor) deposited on the inner surface of the tube, which begins to glow under the influence of the absorbed radiation. The luminescence spectrum, with appropriate selection of phosphors, is close to the spectrum of solar radiation. Glow discharge is used for cathode deposition of metals. The cathode substance in a glow discharge, due to bombardment with positive ions, becomes very hot and goes into a vapor state. By placing various objects near the cathode, they can be coated with a uniform layer of metal.

2. Spark discharge. Occurs under high tension electric field.(≈ 3·10 6 V/m) in a gas under pressure of the order of atmospheric pressure. The spark has the appearance of a brightly glowing thin channel, complexly curved and branched.

The explanation of the spark discharge is given on the basis of the streamer theory, according to which the appearance of a brightly glowing spark channel is preceded by the appearance of faintly glowing accumulations of ionized gas. These clusters are called streamers. Streamers arise not only as a result of the formation of electron avalanches through impact ionization, but also as a result of photon ionization of the gas. Avalanches, catching up with each other, form conducting bridges from streamers, along which at the next moments of time powerful flows of electrons rush, forming spark discharge channels. Due to the release of a large amount of energy during the processes considered, the gas in the spark gap is heated to a very high temperature (approximately 10 4 K), which leads to its glow. Rapid heating of the gas leads to an increase in pressure and the formation of shock waves, which explain the sound effects of a spark discharge - the characteristic crackling sound in weak discharges and powerful thunderclaps in the case of lightning, which is an example of a powerful spark discharge between a thundercloud and the Earth or between two thunderclouds.

A spark discharge is used to ignite a combustible mixture in internal combustion engines and protect electrical transmission lines from overvoltages (spark gaps). When the length of the discharge gap is short, the spark discharge causes destruction (erosion) of the metal surface, so it is used for electric spark precision processing of metals (cutting, drilling). It is used in spectral analysis to register charged particles (spark counters).

3. Arc discharge. If, after igniting a spark discharge from a powerful source, the distance between the electrodes is gradually reduced, then the discharge becomes continuous - an arc discharge occurs. In this case, the current increases sharply, reaching hundreds of amperes, and the voltage across the discharge gap drops to several tens of volts. An arc discharge can be obtained from a low voltage source, bypassing the spark stage. To do this, electrodes (for example, carbon) are brought together until they touch; they become very hot electric shock, then they are separated and an electric arc is obtained (this is how it was discovered by the Russian scientist V.V. Petrov). At atmospheric pressure, the temperature of the cathode is approximately 3900 K. As the arc burns, the carbon cathode becomes sharper, and a depression is formed on the anode - a crater, which is the hottest point of the arc.

According to modern concepts, the arc discharge is maintained due to the high temperature of the cathode due to intense thermionic emission, as well as thermal ionization of molecules caused by high temperature gas

Arc discharge is widely used in national economy for welding and cutting metals, producing high-quality steels (arc furnace), lighting (spotlights, projection equipment). Arc lamps with mercury electrodes in quartz cylinders are also widely used, where an arc discharge occurs in mercury vapor when the air is evacuated. The arc created in mercury vapor is a powerful source of ultraviolet radiation and is used in medicine (for example, quartz lamps). Arc discharge at low pressures in mercury vapor is used in mercury rectifiers to rectify alternating current.

4. Corona discharge – a high-voltage electrical discharge that occurs at high (for example, atmospheric) pressure in a non-uniform field (for example, near electrodes with a large curvature of the surface, the tip of a needle electrode). When the field strength near the tip reaches 30 kV/cm, a glow appears around it in the form of a crown, which gives rise to the name of this type of discharge.

Depending on the sign of the corona electrode, a negative or positive corona is distinguished. In the case of a negative corona, the birth of electrons, causing impact ionization of gas molecules, occurs due to their emission from the cathode under the influence of positive ions, in the case of a positive corona, due to ionization of the gas near the anode. IN natural conditions the corona appears under the influence of atmospheric electricity at the tops of the masts of ships or trees (the action of lightning rods is based on this). This phenomenon was called in ancient times the fires of St. Elmo. Harmful effect corona around the wires of high-voltage power lines is the occurrence of leakage currents. To reduce them, the wires of high-voltage lines are made thick. Corona discharge, being intermittent, also becomes a source of radio interference.

Corona discharge is used in electric precipitators used to purify industrial gases from impurities. The gas to be purified moves from bottom to top in a vertical cylinder, along the axis of which a corona wire is located. Ions present in large quantities in the outer part of the corona, they settle on impurity particles and are carried away by the field to the external non-corona electrode and settle on it. Corona discharge is also used when applying powder and paint coatings.

ELECTROSTATIC FIELD

ELECTRIC FIELD LINES

According to the concepts of modern physics, the effect of one charge on another is transmitted through electrostatic field - a special endlessly extending material environment that every charged body creates around itself. Electrostatic fields cannot be detected by human senses. However, a charge placed in a field is acted upon by a force directly proportional to magnitude this charge. Because the direction of the force depends on the sign of the charge, we agreed to use the so-called test charge q 0. This is a positive, point charge that is placed at the point of the electric field that interests us. Accordingly, as a force characteristic of the field, it is advisable to use the ratio of force to the value of the test charge q 0:

This constant vector quantity for each point of the field equal to the force acting on a unit positive charge is called tension . For the field of a point charge q at a distance r from it:

, (4)

The direction of the vector coincides with the direction of the force acting on the test charge. [E] = N / C or V / m.

In a dielectric medium, the force of interaction between charges, and hence the field strength, decreases by ε times:

, . (5)

When several electrostatic fields are superimposed on each other, the resulting strength is determined as the vector sum of the strengths of each of the fields (superposition principle):

Graphically, the distribution of the electric field in space is depicted using power lines . These lines are drawn so that the tangents to them at any point coincide with. This means that the vector of force acting on the charge, and therefore the vector of its acceleration, also lies on the tangents to the lines of force, which never intersect anywhere. Electrostatic field lines cannot be closed. They start on positive and end on negative charges or go to infinity.

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

· Electric field lines are always perpendicular to the surface of the conductor.

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

20) Let me remind you that these are the energy characteristics of the electric field.

The electric field potential at any point is defined as

.

and is equal to the potential energy of a unit charge introduced into a given point in the field.

If a charge is moved in a field from point 1 to point 2, then a potential difference arises between these points

.

The meaning of potential difference: this is the work of an electric field to move a charge from one point to another.

The field potential can also be interpreted through work. If point 2 is at infinity, where there is no field (), then - this is the work of the field to move a charge from a given point to infinity. The field potential created by a single charge is calculated as .

Surfaces at each point of which the field potentials are the same are called equipotential surfaces. In a dipole field, the potential surfaces are distributed as follows:

The field potential formed by several charges is calculated using the principle of superposition: .

a) Calculation of the potential at point A, located not on the dipole axis:

Let us find from the triangle ( ). Obviously, . That's why And .

.

b) Between points A and B, equidistant from the dipole at a distance

() the potential difference is defined as (we accept without the proof, which you will find in Remizov’s textbook)

.

c) It can be shown that if the dipole is located in the center of an equilateral triangle, then the potential difference between the vertices of the triangle are related as projections of the vector onto the sides of this triangle ( ).

21) - the work of the electric field along the power lines is calculated.

1. Work in an electric field does not depend on the shape of the path.

2. No work is performed perpendicular to the lines of force.

3. In a closed loop, no work is done in an electric field.

Energy characteristics of the electric field (potanceal).

1) Physical meaning:

If Cl, then (numerically), provided that the charge placed at a given point in the electric field.

Unit of measurement:

2) Physical meaning:

If a unit positive point charge is placed at a given point, then (numerically), when moving from a given point to infinity.

Δφ is the difference between the dance values ​​of two points of the electric field.

U – voltage – “y” is the difference between the voltages of two points of the electric field.

[U]=V (Volt)

Physical meaning:

If , then (numerically) when moving from one point of the field to another.

Relationship between stress and tension:

22) In an electrostatic field, all points of a conductor have the same potential, which is proportional to the charge of the conductor, i.e. the ratio of charge q to potential φ does not depend on charge q. (Electrostatic is the field surrounding stationary charges). Therefore, it turned out to be possible to introduce the concept of electrical capacitance C of a solitary conductor:

Electrical capacity is a quantity numerically equal to the charge that must be imparted to the conductor in order for its potential to change by one.

Capacitance is determined by the geometric dimensions of the conductor, its shape and properties environment and does not depend on the conductor material.

Units of measurement for quantities included in the definition of capacity:

Capacitance - designation C, unit of measurement - Farad (F, F);

Electric charge - designation q, unit of measurement - coulomb (C, C);

φ - field potential - volts (V, V).

It is possible to create a system of conductors that will have a capacitance much greater than an individual conductor, independent of the surrounding bodies. Such a system is called a capacitor. The simplest capacitor consists of two conducting plates located at a short distance from each other (Fig. 1.9). The electric field of a capacitor is concentrated between the plates of the capacitor, that is, inside it. Capacitor capacity:

C = q / (φ1 - φ2) = q / U

(φ1 - φ2) - potential difference between the plates of the capacitor, i.e. voltage.

The capacitance of a capacitor depends on its size, shape and dielectric constant ε of the dielectric located between the plates.

C = ε∙εo∙S / d, where

S - lining area;

d - distance between plates;

ε is the dielectric constant of the dielectric between the plates;

εo - electrical constant 8.85∙10-12F/m.

If it is necessary to increase the capacitance, the capacitors are connected to each other in parallel.

Fig.1.10. Parallel connection of capacitors.

Ctotal = C1 + C2 + C3

In a parallel connection, all capacitors are under the same voltage, and their total charge is Q. In this case, each capacitor will receive a charge Q1, Q2, Q3, ...

Q = Q1 + Q2 + Q3

Q1 = C1∙U; Q2 = C2∙U; Q3 = C3∙U. Let's substitute into the above equation:

C∙U = C1∙U + C2∙U + C3∙U, whence C = C1 + C2 + C3 (and so on for any number of capacitors).

For serial connection:

Fig.1.11. Series connection of capacitors.

1/Ctot = 1/C1 + 1/C2 + ∙∙∙∙∙ + 1/ Cn

Derivation of the formula:

Voltage on individual capacitors U1, U2, U3,..., Un. Total voltage of all capacitors:

U = U1 + U2 + ∙∙∙∙∙ + Un,

taking into account that U1 = Q/ C1; U2 = Q/ C2; Un = Q/ Cn, substituting and dividing by Q, we obtain a relationship for calculating the capacitance of a circuit with a series connection of capacitors

Capacitance units:

F - farad. This is a very large value, so smaller values ​​are used:

1 µF = 1 µF = 10-6F (microfarad);

1 nF = 1 nF = 10-9 F (nanofarad);

1 pF = 1pF = 10-12F (picofarad).

23) If a conductor is placed in an electric field then the force q will act on the free charges q in the conductor. As a result, a short-term movement of free charges occurs in the conductor. This process will end when the own electric field of the charges arising on the surface of the conductor completely compensates for the external field. The resulting electrostatic field inside the conductor will be zero (see § 43). However, in conductors, under certain conditions, continuous ordered movement of free electric charge carriers can occur. This movement is called electric current. The direction of the electric current is taken to be the direction of movement of positive free charges. For the existence of electric current in a conductor, two conditions must be met:

1) the presence of free charges in the conductor - current carriers;

2) the presence of an electric field in the conductor.

The quantitative measure of electric current is current strength I– scalar physical quantity equal to the ratio of the charge Δq transferred through the cross section of the conductor (Fig. 11.1) during the time interval Δt to this time interval:

The ordered movement of free current carriers in a conductor is characterized by the speed of the ordered movement of the carriers. This speed is called drift speed current carriers. Let a cylindrical conductor (Fig. 11.1) have a cross section with an area S. In the volume of the conductor, limited by cross sections 1 and 2 with a distance ∆ X between them contains the number of current carriers ∆ N= nS∆X, Where n– concentration of current carriers. Their total charge ∆q = q 0 ∆ N= q 0 nS∆X. If, under the influence of an electric field, current carriers move from left to right with a drift speed v dr, then in time ∆ t=∆x/v dr all carriers contained in this volume will pass through cross section 2 and create an electric current. The current strength is:

. (11.2)

Current density is the amount of electric current flowing through a unit cross-sectional area of ​​a conductor:

. (11.3)

In a metal conductor, the current carriers are free electrons of the metal. Let's find the drift speed of free electrons. With current I = 1A, cross-sectional area of ​​the conductor S= 1mm 2, concentration of free electrons (for example, in copper) n= 8.5·10 28 m --3 and q 0 = e = 1.6·10 –19 C we obtain:

v dr = .

We see that the speed of directed motion of electrons is very low, much less than the speed of chaotic thermal motion of free electrons.

If the strength of the current and its direction do not change over time, then such a current is called constant.

IN International system SI units current is measured in amperes (A). The current unit of 1 A is determined by the magnetic interaction of two parallel conductors with current.

A direct electric current can be created in a closed circuit in which free charge carriers circulate along closed trajectories. But when an electric charge moves in an electrostatic field along a closed path, the work done by electric forces is zero. Therefore, for the existence of direct current, it is necessary to have a device in the electrical circuit that is capable of creating and maintaining potential differences in sections of the circuit due to the work of forces of non-electrostatic origin. Such devices are called direct current sources. Forces of non-electrostatic origin acting on free charge carriers from current sources are called external forces.

The nature of external forces may vary. IN galvanic cells or batteries they arise as a result of electrochemical processes; in direct current generators, external forces arise when conductors move in a magnetic field. Under the influence of external forces, electric charges move inside the current source against the forces of the electrostatic field, due to which a constant electric current can be maintained in a closed circuit.

When electric charges move along a direct current circuit, external forces acting inside the current sources perform work.

Physical quantity equal to the work ratio A st external forces when a charge q moves from the negative pole of a current source to the positive pole to the value of this charge is called the electromotive force of the source (EMF):

ε . (11.2)

Thus, the EMF is determined by the work done by external forces when moving a single positive charge. Electromotive force, like potential difference, is measured in volts (V).

When a single positive charge moves along a closed direct current circuit, the work done by external forces is equal to the sum of the emf acting in this circuit, and the work done by the electrostatic field is zero.

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

More useful tool A visual representation of the vector nature of the electric field is the use of such a concept as, or simply lines of force. Instead of drawing countless vector arrows in space surrounding the source charge, it has proven useful to combine them into lines, where the vectors themselves are tangent to points on such lines.

As a result, they are successfully used to represent the vector picture of the electric field. electric field lines, which come out of charges of a positive sign and enter charges of a negative sign, and also extend to infinity in space. This view allows you to see with your mind what is invisible. to the human eye electric field . However, this representation is also convenient for gravitational forces and any other non-contact long-range interactions.

The model of electrical power lines includes an infinite number of them, but too high density images of field lines reduce the ability to read field patterns, so their number is limited by readability.

Rules for drawing electric field lines

There are many rules for drawing up such models of electrical power lines. All these rules were created in order to provide the greatest information content when visualizing (drawing) the electric field. One way is to depict field lines. One of the most common methods is to surround more charged objects with more lines, that is, with a greater line density. Objects with more charge create stronger electric fields and therefore the density (density) of lines around them is greater. The closer to the charge the source, the higher the density of the lines of force, and the greater the magnitude of the charge, the denser the lines around it.

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

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

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

Electric field consisting of two or more charges

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

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

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

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

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

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

For a visual graphical representation of the field, it is convenient to use lines of force - directed lines, the tangents to which at each point coincide with the direction of the electric field strength vector (Fig. 233).

Rice. 233
According to the definition, electric field lines have a number of general properties(compare with the properties of fluid flow lines):
1. The field lines do not intersect (otherwise, at the point of intersection, two tangents can be constructed, that is, at one point, the field strength has two values, which is absurd).
2. Lines of force do not have breaks (at the break point, two tangents can again be constructed).
3. Electrostatic field lines begin and end at charges.
Since the field strength is determined at each spatial point, the field line can be drawn through any spatial point. Therefore, the number of lines of force is infinitely large. The number of lines that are used to depict the field is most often determined by the artistic taste of the physicist-artist. In some textbooks It is recommended to build a picture of the field lines so that their density is greater where the field strength is greater. This requirement is not strict, and not always feasible, therefore lines of force are drawn, satisfying the formulated properties 1 − 3 .
It is very easy to construct the field lines of the field created by a point charge. In this case, the lines of force are a set of straight lines leaving (for positive) or entering (for negative) the point where the charge is located (Fig. 234).

rice. 234
Such families of field lines of point charge fields demonstrate that the charges are sources of the field, analogous to the sources and sinks of the fluid velocity field. We will prove later that lines of force cannot begin or end at those points where there are no charges.
The picture of field lines of real fields can be reproduced experimentally.
Pour a small layer into a low vessel castor oil and pour a small portion of semolina into it. If the oil and cereal are placed in an electrostatic field, then the grains of semolina (they have a slightly elongated shape) rotate in the direction of the electric field strength and line up approximately along the lines of force; after several tens of seconds, a picture of the electric field lines appears in the cup. Some of these “pictures” are presented in photographs.
It is also possible to carry out theoretical calculations and construction of field lines. True, these calculations require an enormous number of calculations, so they are actually (and without much difficulty) carried out using a computer; most often such constructions are performed in a certain plane.
When developing algorithms for calculating the pattern of field lines, a number of problems are encountered that require resolution. The first such problem is the calculation of the field vector. In the case of electrostatic fields created by a given charge distribution, this problem is solved using Coulomb's law and the principle of superposition. The second problem is the method of constructing a separate line. The idea of ​​the simplest algorithm that solves this problem is quite obvious. In a small area, each line practically coincides with its tangent, so you should construct many segments of tangents to the lines of force, that is, segments of short length l, the direction of which coincides with the direction of the field at a given point. To do this, it is necessary, first of all, to calculate the components of the tension vector at a given point E x, E y and the modulus of this vector E = √(E x 2 + E y 2 ). Then you can construct a short segment, the direction of which coincides with the direction of the field strength vector. its projections on the coordinate axes are calculated using the formulas that follow from Fig. 235:

rice. 235

Then you should repeat the procedure, starting from the end of the constructed segment. Of course, when implementing such an algorithm, there are other problems that are more of a technical nature.
Figures 236 show the field lines created by two point charges.


rice. 236
The signs of the charges are indicated, in figures a) and b) the charges are the same in absolute value, in fig. c), d) are different - we propose to determine which one is better on your own. Also determine the directions of the field lines in each case yourself.
It is interesting to note that M. Faraday considered the electric field lines as real elastic tubes connecting electric charges with each other; such ideas greatly helped him predict and explain many physical phenomena.
Agree that the great M. Faraday was right - if you mentally replace the lines with elastic rubber bands, the nature of the interaction is very clear.

Electric charge is a physical scalar quantity that determines the ability of bodies to be a source of electromagnetic fields and take part in electromagnetic interaction.

In a closed system, the algebraic sum of the charges of all particles remains unchanged.

(... but not the number of charged particles, since there are transformations of elementary particles).

Closed system

- a system of particles into which charged particles do not enter from the outside and do not exit.

Coulomb's law

- the basic law of electrostatics.

The force of interaction between two point stationary charged bodies in a vacuum is directly proportional

the product of the charge modules and is inversely proportional to the square of the distance between them.

When are bodies considered point bodies? - if the distance between them is many times greater than the size of the bodies.

If two bodies have electric charges, then they interact according to Coulomb's law.

Electric field strength. Superposition principle. Calculation of the electrostatic field of a system of pointed charges based on the superposition principle.

Electric field strength is a vector physical quantity that characterizes the electric field at a given point and is numerically equal to the force ratio acting on a stationary [test charge placed at a given point in the field, to the magnitude of this charge :

The principle of superposition is one of the most general laws in many branches of physics. In its simplest formulation, the superposition principle states:

the result of the influence of several external forces on a particle is the vector sum of the influence of these forces.

The best known principle is superposition in electrostatics, in which it states that the strength of the electrostatic field created at a given point by a system of charges is the sum of the field strengths of individual charges.

4. Lines of tension (lines of force) of the electric field. Tension vector flow. Density of power lines.

The electric field is represented using lines of force.

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

Properties of electric field lines

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

Electric field lines are always perpendicular to the surface of the conductor.

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

9.5. Electric field strength vector flux. Gauss's theorem

As with any vector field, it is important to consider the flow properties of the electric field. The electric field flux is defined traditionally.

Let us select a small area with area Δ S, the orientation of which is specified by the unit normal vector (Fig. 157).

Within a small area, the electric field can be considered uniform, then the flux of the intensity vector Δ F E is defined as the product of the site area and the normal component of the tension vector

Where - scalar product of vectors and ; E n is the component of the tension vector normal to the site.

In an arbitrary electrostatic field, the flow of the intensity vector through an arbitrary surface is determined as follows (Fig. 158):

The surface is divided into small areas Δ S(which can be considered flat);

The tension vector on this site is determined (which can be considered constant within the site);

The sum of flows through all areas into which the surface is divided is calculated

This amount is called flow of the electric field strength vector through a given surface.

Continuous lines whose tangents at each point through which they pass coincide with the tension vector are called electric field lines or tension lines.
The density of the lines is greater where the field strength is greater. The lines of force of the electric field created by stationary charges are not closed: they begin on positive charges and end on negative ones. An electric field whose strength is the same at all points in space is called homogeneous. The density of the lines is greater near charged bodies, where the tension is greater. Lines of force of the same field do not intersect. Any charge in an electric field is affected by a force. If a charge moves under the influence of this force, then the electric field does work. The work of forces to move a charge in an electrostatic field does not depend on the trajectory of the charge and is determined only by the position of the starting and ending points. Consider a uniform electric field formed by flat plates charged in opposite ways. The field strength is the same at all points. Let a point charge q move from point A to point B along the curve L. When the charge moves a small amount D L, the work is equal to the product of the magnitude of the force by the magnitude of the displacement and the cosine of the angle between them, or, which is the same, the product of the magnitude of the point charge by the intensity fields and on the projection of the displacement vector onto the direction of the tension vector. If we calculate the total work of moving a charge from point A to point B, then it, regardless of the shape of the curve L, will be equal to the work of moving charge q along the field line to point B 1. The work of moving from point B 1 to point B is zero, since the force vector and the displacement vector are perpendicular.

5. Gauss's theorem for the electric field in vacuum

General wording: Flow vector electric field strength through any arbitrarily chosen closed surface is proportional to what is contained inside this surface electric charge.

GHS	SI

This expression represents Gauss's theorem in integral form.

Comment: the flow of the tension vector through the surface does not depend on the charge distribution (charge arrangement) inside the surface.

In differential form, Gauss's theorem is expressed as follows:

GHS	SI

Here is the volumetric charge density (in the case of the presence of a medium, the total density of free and bound charges), and - obla operator.

Gauss's theorem can be proven as a theorem in electrostatics based on Coulomb's law ( see below). The formula, however, is also true in electrodynamics, although in it it most often does not act as a provable theorem, but acts as a postulated equation (in this sense and context it is more logical to call it Gauss's law .

6. Application of Gauss’s theorem to the calculation of the electrostatic field of a uniformly charged long thread (cylinder)

Field of a uniformly charged infinite cylinder (thread). An infinite cylinder of radius R (Fig. 6) is uniformly charged with linear densityτ (τ = –dQ/dt charge per unit length). From considerations of symmetry, we see that the tension lines will be directed along the radii of the circular sections of the cylinder with equal density in all directions relative to the cylinder axis. Let us mentally construct as a closed surface a coaxial cylinder of radius r and height l. Flow vector E through the ends of the coaxial cylinder is equal to zero (the ends and tension lines are parallel), and through the side surface it is equal to 2πr l E. Using Gauss's theorem, for r>R 2πr l E = τ l/ε 0 , whence (5) If r

7. Application of Gauss's theorem to the calculation of the electrostatic field of a uniformly charged plane

Field of a uniformly charged infinite plane. An infinite plane (Fig. 1) is charged with a constant surface density+σ (σ = dQ/dS - charge per unit surface). The tension lines are perpendicular to this plane and directed from it in each direction. Let us take as a closed surface a cylinder, the bases of which are parallel to the charged plane and the axis is perpendicular to it. Since the generatrices of the cylinder are parallel to the lines of field strength (cosα = 0), the flux of the intensity vector through the side surface of the cylinder is zero, and the total flux through the cylinder is equal to the sum of the fluxes through its bases (the areas of the bases are equal and for the base E n coincides with E), i.e. equal to 2ES. The charge that is contained inside the constructed cylindrical surface is equal to σS. According to Gauss's theorem, 2ES=σS/ε 0, from which (1) From formula (1) it follows that E does not depend on the length of the cylinder, i.e. the field strength at any distance is equal in magnitude, in other words, the field of a uniformly charged plane homogeneously.

8. Application of Gauss's theorem to the calculation of the electrostatic field of a uniformly charged sphere and a volumetrically charged ball.

Field of a uniformly charged spherical surface. A spherical surface of radius R with a total charge Q is charged uniformly with surface density+σ. Because The charge is distributed evenly over the surface; the field that it creates has spherical symmetry. This means that the tension lines are directed radially (Fig. 3). Let us mentally draw a sphere of radius r, which has a common center with a charged sphere. If r>R,ro the entire charge Q gets inside the surface, which creates the field under consideration, and, according to Gauss’s theorem, 4πr 2 E = Q/ε 0, whence (3) For r>R, the field decreases with distance r according to the same law as for a point charge. The dependence of E on r is shown in Fig. 4. If r"

Field of a volumetrically charged ball. A sphere of radius R with total charge Q is charged uniformly with bulk densityρ (ρ = dQ/dV – charge per unit volume). Taking into account symmetry considerations similar to point 3, it can be proven that for the field strength outside the ball the same result will be obtained as in case (3). Inside the ball, the field strength will be different. Sphere of radius r"

9. Work of electric field forces when moving a charge. Theorem on the circulation of electric field strength.

The elementary work done by force F when moving a point electric charge from one point of the electrostatic field to another along a path segment is, by definition, equal to

where is the angle between the force vector F and the direction of movement. If the work is done by external forces, then dA0. Integrating the last expression, we obtain that the work against field forces when moving a test charge from point “a” to point “b” will be equal to

where is the Coulomb force acting on the test charge at each point of the field with intensity E. Then the work

Let a charge move in the field of charge q from point “a”, distant from q at a distance, to point “b”, remote from q at a distance (Fig. 1.12).

As can be seen from the figure, then we get

As mentioned above, the work of electrostatic field forces performed against external forces is equal in magnitude and opposite in sign to the work of external forces, therefore

Electric field circulation theorem.

Tension And potential- these are two characteristics of the same object - the electric field, therefore there must be a functional connection between them. Indeed, the work of field forces to move a charge q from one point in space to another can be represented in two ways:

Whence it follows that

This is what you are looking for connection between the intensity and potential of the electric field in differential form.

- a vector directed from a point with less potential to a point with greater potential (Fig. 2.11).

, .

Fig.2.11. Vectors And gradφ. .

From the property of potentiality of the electrostatic field it follows that the work of field forces along a closed loop (φ 1 = φ 2) is equal to zero:

so we can write

The last equality reflects the essence second main theorem electrostatics – electric field circulation theorems , according to which field circulation along of an arbitrary closed contour is equal to zero. This theorem is a direct consequence potentiality electrostatic field.

10. Electric field potential. The relationship between potential and tension.

Electrostatic potential(see also Coulomb potential ) - scalar energy characteristic electrostatic field, characterizing potential energy field that a single charge, placed at a given point in the field. Unit of measurement potential is therefore a unit of measurement work, divided by unit of measurement charge(for any system of units; more about units of measurement - see below).

Electrostatic potential- a special term for a possible replacement for the general term of electrodynamics scalar potential in a special case electrostatics(historically, the electrostatic potential appeared first, and the scalar potential of electrodynamics is its generalization). Use of the term electrostatic potential determines the presence of an electrostatic context. If such a context is already obvious, they often simply talk about potential without qualifying adjectives.

The electrostatic potential is equal to the ratio potential energy interaction charge with the field to the magnitude of this charge:

Electrostatic field strength and potential are related by the relation

or vice versa :

Here - obla operator , that is, on the right side of the equality there is a minus gradient potential - a vector with components equal to partial derivative from the potential along the corresponding (rectangular) Cartesian coordinates, taken with the opposite sign.

Using this relation and Gauss's theorem for the field strength, it is easy to see that the electrostatic potential satisfies Poisson's equation. In system units SI:

where is the electrostatic potential (in volts), - volumetric charge density(V pendants per cubic meter), a - vacuum (in farads per meter).

11. Energy of a system of stationary point electric charges.

Energy of a system of stationary point charges. As we already know, electrostatic interaction forces are conservative; This means that the system of charges has potential energy. We will look for the potential energy of a system of two stationary point charges Q 1 and Q 2, which are located at a distance r from each other. Each of these charges in the field of the other has potential energy (we use the formula for the potential of a solitary charge): where φ 12 and φ 21 are, respectively, the potentials that are created by the charge Q 2 at the point where the charge Q 1 is located and by the charge Q 1 at the point where the charge Q 2 is located. According to, and therefore W 1 = W 2 = W and By adding charges Q 3, Q 4, ... to our system of two charges in succession, we can prove that in the case of n stationary charges, the interaction energy of the system of point charges is equal to (1) where φ i is the potential that is created at the point where the charge Q i is located by all charges except the i-th one.

12. Dipole in an electric field. Polar and non-polar molecules. Polarization of dielectrics. Polarization. Ferroelectrics.

If you place a dielectric in an external electric field, it will become polarized, i.e., it will receive a non-zero dipole moment pV = ∑pi, where pi is the dipole moment of one molecule. To produce a quantitative description of the polarization of a dielectric, a vector quantity is introduced - polarization, which is defined as the dipole moment per unit volume of the dielectric:

It is known from experience that for a large class of dielectrics (with the exception of ferroelectrics, see below) the polarization P depends linearly on the field strength E. If the dielectric is isotropic and E is not numerically too large, then

Ferroelectrics- dielectrics that have spontaneous (spontaneous) polarization in a certain temperature range, that is, polarization in the absence of an external electric field. Ferroelectrics include, for example, the Rochelle salt NaKC 4 H 4 O 6 4H 2 O, studied in detail by I. V. Kurchatov (1903-1960) and P. P. Kobeko (1897-1954) (from which this name was derived) and Barium titanate BaTiO 3 .

Polarization of dielectrics- a phenomenon associated with limited displacement of associated charges V dielectric or by turning electric dipoles, usually under the influence of external electric field, sometimes under the influence of other external forces or spontaneously.

The polarization of dielectrics is characterized by electric polarization vector . The physical meaning of the electric polarization vector is dipole moment, per unit volume of the dielectric. Sometimes the polarization vector is briefly called simply polarization.

Electric dipole- an idealized electrically neutral system consisting of point and equal in absolute value positive and negative electric charges.

In other words, an electric dipole is a combination of two equal in absolute value opposite point charges located at a certain distance from each other

The product of the vector conducted from a negative charge to a positive one by the absolute value of the charges is called the dipole moment:

In an external electric field, a moment of force acts on an electric dipole, which tends to rotate it so that the dipole moment turns along the direction of the field.

The potential energy of an electric dipole in a (constant) electric field is equal to (In the case of a non-uniform field, this means dependence not only on the moment of the dipole - its magnitude and direction, but also on the location, the point of location of the dipole).

Far from the electric dipole, its intensity electric field decreases with distance, that is, faster than point charge ().

Any generally electrically neutral system containing electric charges, in some approximation (that is, actually in dipole approximation) can be considered as an electric dipole with a moment where is the charge of the th element and is its radius vector. In this case, the dipole approximation will be correct if the distance at which the electric field of the system is studied is large compared to its characteristic dimensions.

Polar substances V chemistry - substances, molecules which they have electric dipole moment. Polar substances, in comparison with non-polar ones, are characterized by high the dielectric constant(more than 10 in the liquid phase), increased boiling temperature And melting temperature.

The dipole moment usually arises due to different electronegativity components of the molecule atoms, because of which communications in the molecule acquire polarity. However, to acquire a dipole moment, not only the polarity of the bonds is required, but also their corresponding location in space. Molecules having a shape similar to molecules methane or carbon dioxide, are non-polar.

Polar solvents most willingly dissolve polar substances, and also have the ability solvate ions. Examples of a polar solvent are water, alcohols and other substances.

13. Electric field strength in dielectrics. Electrical bias. Gauss's theorem for the field in dielectrics.

The electrostatic field strength, according to (88.5), depends on the properties of the medium: in a homogeneous isotropic medium, the field strength E is inversely proportional to . Tension vector E, passing through the boundary of dielectrics, undergoes an abrupt change, thereby creating inconvenience when calculating electrostatic fields. Therefore, it turned out to be necessary, in addition to the intensity vector, to characterize the field electric displacement vector, which for an electrically isotropic medium, by definition, is equal to

Using formulas (88.6) and (88.2), the electric displacement vector can be expressed as

The unit of electrical displacement is coulomb per meter squared (C/m2).

Let's consider what the electric displacement vector can be associated with. Bound charges appear in a dielectric in the presence of an external electrostatic field created by a system of free electric charges, i.e. in a dielectric an additional field of bound charges is superimposed on the electrostatic field of free charges. Result field in a dielectric is described by the voltage vector E, and therefore it depends on the properties of the dielectric. Vector D describes the electrostatic field created free charges. Bound charges arising in the dielectric can, however, cause a redistribution of free charges that create the field. Therefore the vector D characterizes the electrostatic field created free charges(i.e. in a vacuum), but with such distribution in space as there is in the presence of a dielectric.

Same as the field E, field D depicted using electric displacement lines, the direction and density of which are determined in exactly the same way as for tension lines (see §79).

Vector lines E can begin and end on any charges - free and bound, while vector lines D - only on free charges. Through the field areas where the bound charges are located, the vector lines D pass without interruption.

For free closed surfaces S vector flow D through this surface

Where D n- vector projection D to normal n to site d S.

Gauss's theorem For electrostatic field in a dielectric:

(89.3)

i.e., the flux of the displacement vector of the electrostatic field in a dielectric through an arbitrary closed surface is equal to the algebraic sum of those contained within this surface free electric charges. In this form, Gauss's theorem is valid for the electrostatic field both for homogeneous and isotropic and for inhomogeneous and anisotropic media.

For vacuum D n =  0 E n ( =1), then the flux of the tension vector E through an arbitrary closed surface (cf. (81.2)) is equal to

Since the field sources E there are both free and bound charges in the medium, then the Gauss theorem (81.2) for the field E in the most general form can be written as

where are, respectively, the algebraic sums of free and bound charges covered by a closed surface S. However, this formula is unacceptable for describing the field E in a dielectric, since it expresses the properties of an unknown field E through associated charges, which, in turn, are determined by it. This once again proves the feasibility of introducing an electrical displacement vector.

. Electric field strength in a dielectric.

In accordance with superposition principle The electric field in a dielectric is a vectorial sum of the external field and the field of polarization charges (Fig. 3.11).

or by absolute value

We see that the field strength in a dielectric is less than in a vacuum. In other words, any dielectric weakens external electric field.

Fig.3.11. Electric field in a dielectric.

Electric field induction , where , , that is . On the other hand, whence we find that ε 0 E 0 = ε 0 εE and, therefore, the electric field strength in isotropic dielectric has:

This formula reveals physical meaning dielectric constant and shows that the electric field strength in the dielectric is times less than in a vacuum. This leads to a simple rule: to write the formulas of electrostatics in a dielectric, it is necessary in the corresponding formulas of vacuum electrostatics next to attribute .

In particular, Coulomb's law in scalar form will be written as:

14. Electric capacity. Capacitors (flat, spherical, cylindrical), their capacities.

A capacitor consists of two conductors (plates) that are separated by a dielectric. The capacitance of the capacitor should not be affected by surrounding bodies, therefore the conductors are shaped in such a way that the field created by the accumulated charges is concentrated in a narrow gap between the plates of the capacitor. This condition is satisfied by: 1) two flat plates; 2) two concentric spheres; 3) two coaxial cylinders. Therefore, depending on the shape of the plates, capacitors are divided into flat, spherical and cylindrical.

Since the field is concentrated inside the capacitor, the intensity lines begin on one plate and end on the other, therefore the free charges that arise on different plates are equal in magnitude and opposite in sign. Under capacity capacitor is understood as a physical quantity equal to the ratio of the charge Q accumulated in the capacitor to the potential difference (φ 1 - φ 2) between its plates: (1) Let us find the capacitance of a flat capacitor, which consists of two parallel metal plates of area S each, located at a distance d from each other and having charges +Q and –Q. If we assume that the distance between the plates is small compared to their linear dimensions, then the edge effects on the plates can be neglected and the field between the plates can be considered uniform. It can be found using the formula for the field potential of two infinite parallel oppositely charged planes φ 1 -φ 2 =σd/ε 0. Considering the presence of a dielectric between the plates: (2) where ε is the dielectric constant. Then from formula (1), replacing Q=σS, taking into account (2), we find an expression for the capacitance of a flat capacitor: (3) To determine the capacitance of a cylindrical capacitor, which consists of two hollow coaxial cylinders with radii r 1 and r 2 (r 2 > r 1), one is inserted into the other, again neglecting edge effects, we consider the field to be radially symmetric and acting only between the cylindrical plates. We calculate the potential difference between the plates using the formula for the potential difference in the field of a uniformly charged infinite cylinder with linear density τ =Q/ l (l- length of the linings). If there is a dielectric between the plates, the potential difference is (4) Substituting (4) into (1), we find an expression for the capacitance of a cylindrical capacitor: (5) To find the capacitance of a spherical capacitor, which consists of two concentric plates separated by a spherical layer of dielectric, we use the formula for potential difference between two points lying at distances r 1 and r 2 (r 2 > r 1) from the center of the charged spherical surface. If there is a dielectric between the plates, the potential difference (6) Substituting (6) into (1), we obtain

Electrical capacity- characteristic of a conductor, a measure of its ability to accumulate electric charge. In electrical circuit theory, capacitance is the mutual capacitance between two conductors; parameter of a capacitive element of an electrical circuit, presented in the form of a two-terminal network. This capacity is defined as the ratio of the magnitude of the electric charge to potential difference between these conductors.

In system SI capacity is measured in farads. In system GHS V centimeters.

For a single conductor, capacitance is equal to the ratio of the conductor's charge to its potential, assuming that all other conductors endlessly removed and that the potential of the point at infinity is taken to be zero. In mathematical form, this definition has the form

Where - charge, - conductor potential.

Capacitance is determined by the geometric dimensions and shape of the conductor and the electrical properties of the environment (its dielectric constant) and does not depend on the material of the conductor. For example, the capacity of a conducting ball of radius R equal (in SI system):

Where ε 0 - electrical constant, ε - .

The concept of capacitance also refers to a system of conductors, in particular, to a system of two conductors separated dielectric or vacuum, - To capacitor. In this case mutual capacitance of these conductors (capacitor plates) will be equal to the ratio of the charge accumulated by the capacitor to the potential difference between the plates. For a parallel plate capacitor the capacitance is equal to:

Where S- area of ​​one plate (it is assumed that they are equal), d- distance between the plates, ε - relative dielectric constant environment between the plates, ε 0 = 8.854·10 −12 F/m - electrical constant.

Capacitor(from lat. condensare- “compact”, “thicken”) - two-terminal network with a certain meaning containers and low ohmic conductivity; storage device charge and electric field energy. A capacitor is a passive electronic component. Typically consists of two plate-shaped electrodes (called linings), separated dielectric, the thickness of which is small compared to the size of the plates.

15. Connection of capacitors (parallel and series)

In addition to what is shown in Fig. 60 and 61, as well as in Fig. 62, and for a parallel connection of capacitors, in which all positive and all negative plates are connected to each other, sometimes the capacitors are connected in series, i.e., so that the negative plate Rice. 62. Connection of capacitors: a) parallel; b) sequential the first capacitor was connected to the positive plate of the second, the negative plate of the second to the positive plate of the third, etc. (Fig. 62, b). In the case of a parallel connection, all capacitors are charged to the same potential difference U, but the charges on them may be different. If their capacitances are equal to C1, C2,..., Cn, then the corresponding charges will be The total charge on all capacitors and, therefore, the capacitance of the entire system of capacitors (35.1) So, the capacitance of a group of parallel-connected capacitors is equal to the sum of the capacitances of individual capacitors. In the case of series-connected capacitors (Fig. 62, b), the charges on all capacitors are equal. Indeed, if we place, for example, a charge +q on the left plate of the first capacitor, then due to induction a charge -q will appear on its right plate, and a charge +q will appear on the left plate of the second capacitor. The presence of this charge on the left plate of the second capacitor, again due to induction, creates a charge -q on its right plate, and a charge +q on the left plate of the third capacitor, etc. Thus, the charge of each of the series-connected capacitors is equal to q. The voltage on each of these capacitors is determined by the capacitance of the corresponding capacitor: where Ci is the capacitance of one capacitor. The total voltage between the outer (free) plates of the entire group of capacitors. Therefore, the capacitance of the entire capacitor system is determined by the expression (35.2) From this formula it is clear that the capacitance of a group of series-connected capacitors is always less than the capacitance of each of these capacitors individually.

16. Electric field energy and its volumetric density.

Electric field energy. The energy of a charged capacitor can be expressed in terms of quantities characterizing the electric field in the gap between the plates. Let's do this using the example of a flat capacitor. Substituting the expression for capacitance into the formula for capacitor energy gives

Private U / d equal to the field strength in the gap; work S· d represents the volume V occupied by the field. Hence,

If the field is uniform (which is the case in a flat capacitor at a distance d much smaller than the linear dimensions of the plates), then the energy contained in it is distributed in space with a constant density w. Then volumetric energy density electric field is equal

Taking into account the relationship, we can write

In an isotropic dielectric, the directions of the vectors D And E coincide and Substitute the expression , we get

The first term in this expression coincides with the field energy density in vacuum. The second term represents the energy spent on polarization of the dielectric. Let us demonstrate this using the example of a non-polar dielectric. The polarization of a non-polar dielectric is that the charges that make up the molecules are displaced from their positions under the influence of an electric field E. Per unit volume of dielectric, work expended on displacing charges q i by value d r i, is

The expression in parentheses is the dipole moment per unit volume or the polarization of the dielectric R. Hence, . Vector P associated with a vector E ratio Substituting this expression into the formula for work, we get

Having carried out the integration, we determine the work spent on the polarization of a unit volume of the dielectric

Knowing the field energy density at each point, you can find the field energy contained in any volume V. To do this you need to calculate the integral:

17. Direct electric current, its characteristics and conditions of existence. Ohm's law for a homogeneous section of a circuit (integral and differential forms)

For the existence of a constant electric current, the presence of free charged particles and the presence of a current source are necessary. in which any type of energy is converted into the energy of an electric field.

Current source - a device in which any type of energy is converted into the energy of an electric field. In a current source, external forces act on charged particles in a closed circuit. The reasons for the occurrence of external forces in different current sources are different. For example, in batteries and galvanic cells, external forces arise due to the occurrence of chemical reactions, in power plant generators they arise when a conductor moves in a magnetic field, in photocells - when light acts on electrons in metals and semiconductors.

Electromotive force of the current source is the ratio of the work of external forces to the amount of positive charge transferred from the negative pole of the current source to the positive one.