Electrostatics, a section of the theory of electricity, which studies the interaction of stationary electric charges. At the heart of electrostatics, which studies the stationary force interaction between macroscopic immobile charged bodies, there are three experimentally established facts: the presence of two types of electric charges, the existence of interaction between them, carried out by an electric field, and the principle of superposition, when the interaction of any two charges does not depend on the presence of others.
There are two types of charges, positive, denoted by a plus sign "+", and negative, which are assigned a minus sign "-". The charges create an electric field around them. The field of stationary charges is an electrostatic field. Electric charge and electric field are the primary concepts of electrostatics.
The total charge of the body, both positive and negative, is always a multiple of some elementary electric charge. In electrostatics, physical quantities averaged in space and time are studied. When averaging in space, the usual methods of physics of continuous media are used, averaging over time allows us to consider stationary charges in thermal motion. Positive and negative charges are constituent parts of molecules, and all macroscopic bodies contain a huge amount of positive and negative charges, but electrostatic interaction is spoken of only when the body has an excess of charges of the same sign. The charge of a macroscopic body is determined by the total charge of elementary particles that make up this body. Averaging makes it possible to consider not only individual charges, but also to introduce an idea of the volumetric charge density. The law of conservation of charges states that in a closed system, charge is conserved.
The measure of the electric field, carrying out the interaction of charges, at any point is the strength. An electric field is depicted using lines of force - lines whose tangent coincides with the direction of the field strength. The field strength at any point is proportional to the magnitude of the generating charge, therefore, in principle, it is possible to associate a certain limited number of lines of force with an elementary charge.
Electric charges of the same sign repel each other, charges of the opposite sign attract. The principle of operation of the electrometer is based on this phenomenon. Registration of the interaction of charges is always carried out at distances much larger than interatomic. Between the electric charges, the size of which can be neglected, a force acts, the magnitude of which is determined by Coulomb's law. Coulomb's law - the basic law of electrostatics, determines the force of interaction of stationary point charges, depending on their magnitude and the distance between them.
It follows from Coulomb's law that the work of electrical forces when moving a charge does not depend on the path along which the charge moves from one point to another, but is determined only by the position of these points in space. If one of the points is carried away to infinity, then at each point it is possible to associate an electric potential that characterizes the work that needs to be done in order to transfer a unit charge from infinity to a given point. If we connect all points with the same potential in an electric field, then we get a surface of equal potentials, or an equipotential surface.
The principle of superposition of electric fields is one of the basic principles of electrostatics, and is a generalization of many observations. In accordance with the principle of superposition, the strength of the electrical E fields of several stationary point charges q1, q2, q3... is equal to the vector sum of the field strengths that each of these charges would create in the absence of the others. In fact, it means that the presence of other charges does not affect the field generated by a given charge.
The law of interaction of electric charges can be formulated in the form of Gauss's theorem, which can be considered as a consequence of Coulomb's law and the principle of superposition. Typical problems of electrostatics are finding the distribution of charges on the surfaces of conductors by the known total charges or potentials of each of them, as well as calculating the energy of the system of conductors by their charges and potentials. Electrostatics also studies the behavior of various materials - conductors and dielectrics - in an electric field.
... All predictions of electrostatics follow from its two laws.
But it's one thing to say these things mathematically, and quite another -
apply them with ease and with 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. With the discovery of electromagnetic phenomena and the revolution in technology that they produced, interest in electrostatics was lost for a while. However, modern scientific research shows the great 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 primordial atmosphere of the Earth, consisting of methane, ammonia, hydrogen and vapors water. Over the next 50 years, other researchers repeated these experiments and received the same results. When short pulses of current are passed through bacteria, pores appear in their envelope (membrane) through which DNA fragments of other bacteria can pass inside, triggering one of the evolutionary mechanisms. Thus, the energy required for the origin of life on Earth and its evolution could really be the electrostatic energy of lightning discharges (Fig. 1).
How electrostatics cause lightning
At each moment of time, about 2000 lightning flashes at different points on the Earth, about 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 to the Earth negative charge. In this case, each of the discharges supplies the Earth with several tens of coulombs of electricity, and the amplitude of the current during a lightning strike is 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.
With the help of measuring instruments installed on atmospheric probes, at the beginning of the 20th century, the electric field of the Earth was measured, the intensity of which at the surface turned out to be approximately 100 V / m, which corresponds to the total charge of the planet of about 400,000 C. The charge carriers in the Earth's atmosphere are ions, the concentration of which increases with height and reaches a maximum at an altitude of 50 km, where an electrically conductive layer, the ionosphere, was formed under the action of cosmic radiation. Therefore, we can say that the electric field of the Earth is the field of a spherical capacitor with an applied voltage of about 400 kV. Under the influence of this voltage, a current of 2–4 kA flows continuously from the upper layers to the lower ones, 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 electrical capacitor of the Earth is discharged, and during a thunderstorm it is charged.
A thundercloud is a huge amount of steam, part 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 pieces of ice of different sizes, since the temperature there is always below zero. These pieces of ice are in constant motion, caused by the ascending currents of warm air rising from the bottom of the heated surface of the earth. Small pieces of ice are lighter than large ones, and they are carried away by ascending air currents and all the time collide with large ones on the way. With each such collision, electrification occurs, in which large ice floes are charged negatively, and small ones positively. Over time, positively charged small ice floes gather 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 is negatively charged. In this case, positive charges are induced on the ground directly under the thundercloud. Now everything is ready for a lightning discharge, in which air breakdown occurs and a negative charge from the bottom of the thundercloud flows to the Earth.
It is characteristic that before a thunderstorm, the strength of the Earth's electric field can reach 100 kV / m, that is, 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 number of times, and they, pushing away from each other, stand on end (Fig. 3).
Fulgurite - lightning trail on the ground
During a lightning discharge, energy is released on the order of 10 9 –10 10 J. Most of this energy is spent on thunder, heating the air, a flash of light and the emission 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 quite enough to cause a fire, kill a person or destroy a building. Lightning can heat the channel through which it travels up to 30,000 ° C, which is much higher than the melting point of sand (1600-2000 ° C). Therefore, lightning, falling into the sand, melts it, and the hot air and water vapor, expanding, form a tube from the molten sand, which solidifies after a while. This is how fulgurites (thunder arrows, devil's fingers) are born - hollow cylinders made of melted sand (Fig. 4). The longest of the 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 threaten human health. However, it is believed that over a thousand people around the world are killed by lightning every year. At least in the United States, 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), Peter van Muschenbroek, in an article on electricity written for the famous French Encyclopedia, defended the traditional methods of preventing lightning - bell ringing and cannon firing, which he believed were quite effective.
In 1750, Franklin invented the lightning rod (lightning rod). In an attempt to protect the Maryland State Capitol building from a lightning strike, he attached a thick iron rod to the building, towering several meters above the dome and connected to the ground. The scientist refused to patent his invention, wishing that it began to serve people as soon as possible. The mechanism of action of a lightning rod can be easily explained if we remember that the electric field strength near the surface of a charged conductor increases with an increase in the 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 a corona discharge in it. As a result, the probability of lightning striking a lightning rod will significantly increase. So knowledge of electrostatics not only allowed us to explain the origin of lightning, but also to find a way to protect ourselves from 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 broke lightning rods for pious reasons.
A curious case occurred in 1780 in a small town in the north of France, where the townspeople demanded to demolish the iron mast of the lightning rod and the matter came to trial. The young lawyer who defended the lightning rod from the attacks of obscurantists built 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 to save lives is for the good, the young lawyer argued. He won the trial and became very famous. 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 100-dollar bill.
Life-giving electrostatics
The capacitor discharge energy not only led to the emergence of life on Earth, but can also restore life to people whose heart cells have ceased to contract synchronously. Asynchronous (chaotic) contraction of heart cells is called fibrillation. Fibrillation of the heart can be stopped if a short current pulse is passed 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 energy of the discharge through the chest can reach 400 J (which is equal to the potential energy of a pound weight raised to a height of 2.5 m). A device that provides an electrical shock that stops fibrillation of the heart is called a defibrillator. The simplest defibrillator is an oscillatory circuit consisting of a 20 μF capacitor and a 0.4 H coil. 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 Ohm, you can get the current pulse necessary to return the patient to life.
Electrostatics giving light
A fluorescent lamp can serve as a convenient indicator of the strength of the electric field. To verify this, while in a dark room, rub the lamp with a towel or a scarf - as a result, the outer surface of the lamp glass will be charged positively, and the fabric will be negatively charged. As soon as this happens, we will see flashes of light arising in those places of the lamp, to which we touch the charged tissue. Measurements have shown that the electric field strength inside a working fluorescent lamp is about 10 V / m. With this intensity, free electrons have the necessary energy to ionize the mercury atoms inside the fluorescent lamp.
The electric field under high-voltage power lines - transmission lines - can reach very high values. Therefore, if at night a fluorescent lamp is stuck into the ground under a power transmission line, then it will light up, and quite brightly (Fig. 5). So with the help of the energy of the electrostatic field, you can illuminate the space under the power line.
How electrostatics warn of fire and make smoke cleaner
In most cases, when choosing a type of fire alarm detector, preference is given to a smoke detector, since a fire is usually accompanied by the release of a large amount of smoke and it is this type of detector that is able to warn people in a building about the danger. Smoke detectors use ionization, or the photoelectric principle, to detect smoke in the air.
In ionization smoke detectors there is a source of α-radiation (usually americium-241), ionizing air between metal plates-electrodes, the 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, to which the electrical circuit reacts, giving an alarm signal. Sensors based on this principle demonstrate a very impressive sensitivity, reacting even before the very first sign of smoke is detected by a living being. 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 electrifying power of dust particles 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 living in our house. Studies have shown that asthma attacks are triggered by one of the proteins 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, such as a Chizhevsky chandelier (Fig. 7). At the same time, the frequency of asthma attacks decreases.
Electrostatics helps not only to neutralize the proteins secreted by insects, but also to catch them. It has already been said that the hair "stands on end" if it is charged. One can imagine what insects feel when they are electrically charged. The thinnest hairs on their paws diverge in different directions, and insects lose their ability to move. The cockroach trap shown in Figure 8 is based on this principle. Cockroaches are attracted by a sweet powder, previously electrostatically charged. Powder (in the figure it is white) 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 just with jets of air. In everyday life and at work, the charges that arise in this way are often called static electricity.
Under normal atmospheric conditions, natural fibers (from cotton, wool, silk and viscose) absorb moisture well (hydrophilic) and therefore conduct electricity slightly. When such fibers touch or rub against other materials, excess electrical charges appear on their surfaces, but for a very short time, since the charges immediately flow back down the wet fabric fibers containing various ions.
Unlike natural fibers, synthetic fibers (polyester, acrylic, polypropylene) poorly absorb moisture (hydrophobic), and there are fewer mobile ions on their surfaces. When synthetic materials come into contact 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 discomfort. By the way, hair is very close in structure to synthetic fibers and is also hydrophobic, therefore, upon contact, for example, with a comb, they are charged with electricity and begin to repel each other.
To get rid of static electricity, the surface of clothing or other object can be smeared with a substance that retains moisture and thereby increases the concentration of mobile ions on the surface. After such treatment, the resulting electric 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, the molecules of which are similar to soap molecules - one part of a very long molecule is charged and the other is not. Substances that prevent 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 soot is included in the impregnation of carpets and upholstery materials. For the same purposes, up to 3% of natural fibers, and sometimes thin metal threads, are added to such materials.
Electrostatics- this is a branch of physics, where the properties and interactions of electrically charged bodies or particles that are motionless relative to the inertial frame of reference are studied, or particles that have an electric charge.
Electric charge is a physical quantity that characterizes the property of bodies or particles to enter into electromagnetic interactions and determines the values of forces and energies during these interactions. In the International System of Units, the unit for measuring electric charge is the coulomb (Cl).
There are two types of electric charges:
- positive;
- negative.
A body is electrically neutral if the total charge of the negatively charged particles that make up the body is equal to the total charge of the positively charged particles.
Elementary particles and antiparticles are stable carriers of electric charges.
The carriers of a positive charge are a proton and a positron, and a negative charge is an electron and an antiproton.
The total electric charge of the system is equal to the algebraic sum of the charges of the bodies included in the system, i.e.:
Charge conservation law: in a closed, electrically isolated system, the total electric charge remains unchanged, no matter what processes occur inside the system.
Isolated system- this is a system into which electrically charged particles or any bodies do not penetrate from the external environment through its boundaries.
Charge conservation law- this is a consequence of the conservation of the number of particles, a redistribution of particles in space occurs.
Conductors- these are bodies that have electric charges that can move freely over considerable distances.
Examples of conductors: metals in solid and liquid states, ionized gases, electrolyte solutions.
Dielectrics- these are bodies that have charges that cannot move from one part of the body to another, i.e., bound charges.
Examples of dielectrics: quartz, amber, ebonite, gases under normal conditions.
Electrification- this is such a process as a result of which bodies acquire the ability to take part in electromagnetic interaction, that is, they acquire an electric charge.
Electrifying bodies- this is a process of redistribution of electric charges in bodies, as a result of which the charges of bodies become of opposite signs.
Types of electrification:
- Electrification due to electrical conductivity... When two metal bodies touch, one charged and the other neutral, then a certain number of free electrons from the charged body to a neutral one, if the charge of the body was negative, and vice versa, if the charge of the body is positive.
As a result of this, in the first case, the neutral body will receive a negative charge, in the second - a positive one.
- Friction electrification... As a result of contact during friction of some neutral bodies, electrons are transferred from one body to another. Frictional electrification is a cause of static electricity, which can be noticed when, for example, brushing your hair with a plastic comb or removing your synthetic shirt or sweater.
- Electrification through influence arises if a charged body is brought to the end of a neutral metal rod, while a violation of the uniform distribution of positive and negative charges occurs in it. Their distribution occurs in a peculiar way: in one part of the rod, an excess negative charge arises, and in the other - a positive one. Such charges are called induced, the occurrence of which is explained by the movement of free electrons in the metal under the action of the electric field of a charged body brought to it.
Point charge is a charged body, the dimensions of which can be neglected under these conditions.
Point charge is a material point that has an electric charge.
Charged bodies interact with each other in the following way: oppositely charged ones attract, like charged ones repel.
Coulomb's law: the force of interaction of two stationary point charges q1 and q2 in vacuum is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them:
The main property of the electric field- this is that the electric field affects electric charges with some force. An electric field is a special case of an electromagnetic field.
Electrostatic field is the electric field of stationary charges. Electric field strength is a vector quantity that characterizes the electric field at a given point. The field strength at a given point is determined by the ratio of the force acting on a point charge placed at a given point of the field to the value of this charge:
Tension is the power characteristic of the electric field; it allows you to calculate the force acting on this charge: F = qE.
In the International System of Units, the unit for measuring strength is volt per meter. Strength lines are imaginary lines needed to use a graphical representation of an electric field. The lines of tension are drawn so that the tangents to them at each point in space coincide in direction with the vector of the field strength at this point.
The principle of superposition of fields: the field strength from several sources is equal to the vector sum of the field strengths of each of them.
Electric dipole is a set of two equal in magnitude unlike point charges (+ q and –q), located at some distance from each other.
Dipole (electrical) moment is a vector physical quantity that is the main characteristic of a dipole.
In the International System of Units, the unit for measuring the dipole moment is coulomb-meter (C / m).
Dielectric types:
- Polar, which include molecules in which the centers of distribution of positive and negative charges do not coincide (electric dipoles).
- Non-polar, in molecules and atoms of which the centers of distribution of positive and negative charges coincide.
Polarization is a process that occurs when dielectrics are placed in an electric field.
Dielectric polarization- This is the process of displacement of the associated positive and negative charges of the dielectric in opposite directions under the action of an external electric field.
The dielectric constant is a physical quantity that characterizes the electrical properties of a dielectric and is determined by the ratio of the modulus of the strength of the electric field in vacuum to the modulus of the strength of this field inside a homogeneous dielectric.
Dielectric constant is a dimensionless quantity and is expressed in dimensionless units.
Ferroelectrics is a group of crystalline dielectrics that do not have an external electric field and instead of it there is a spontaneous orientation of the dipole moments of particles.
Piezoelectric effect- this is the effect of mechanical deformation of some crystals in certain directions, where opposite electric charges appear on their faces.
Electric field potential. Electric capacity
Potential electrostatic is a physical quantity that characterizes the electrostatic field at a given point, it is determined by the ratio of the potential energy of interaction of a charge with a field to the value of a charge placed at a given point of the field: 
In the International System, the unit of measurement is the volt (V).
The field potential of a point charge is determined by: 
Under the conditions if q> 0, then k> 0; if q
The principle of superposition of fields for a potential: if an electrostatic field is created by several sources, then its potential at a given point in space is defined as the algebraic sum of potentials:
The potential difference between two points of an electric field is a physical quantity determined by the ratio of the work of electrostatic forces to move a positive charge from the initial point to the final point to this charge: 
Equipotential surfaces is the geometrical area of points of the electrostatic field, where the potential values are the same.
Electric capacity is a physical quantity that characterizes the electrical properties of a conductor, a quantitative measure of its ability to hold an electric charge.
The electrical capacitance of a solitary conductor is determined by the ratio of the conductor's charge to its potential, while we assume that the field potential of the conductor is taken equal to zero at the infinitely distant point:
Ohm's law
Homogeneous chain section is a section of a circuit that does not have a current source. The voltage in such a section will be determined by the potential difference at its ends, i.e.: 
In 1826, the German scientist G. Ohm discovered a law that determines the relationship between the current in a homogeneous section of the circuit and the voltage on it: the current in a conductor is directly proportional to the voltage on it. , where G is the coefficient of proportionality, which is called in this law the electrical conductivity or the conductivity of the conductor, which is determined by the formula.
Conductivity of a conductor is a physical quantity that is the reciprocal of its resistance.
In the International System of Units, the unit for measuring electrical conductivity is Siemens (Cm).
The physical meaning of Siemens: 1 cm is the conductivity of a conductor with a resistance of 1 ohm.
To obtain Ohm's law for a section of a circuit, it is necessary to substitute the resistance R instead of electrical conductivity in the formula above, then:
Ohm's law for a section of a chain: the current in the circuit section is directly proportional to the voltage on it and inversely proportional to the resistance of the circuit section.
Ohm's law for a complete circuit: the current strength in an unbranched closed circuit, including a current source, is directly proportional to the electromotive force of this source and is inversely proportional to the sum of the external and internal resistances of this circuit:
Sign rules:
- If, when bypassing the circuit in the selected direction, the current inside the source goes in the direction of the bypass, then the EMF of this source is considered positive.
- If, when bypassing the circuit in the selected direction, the current inside the source goes in the opposite direction, then the EMF of this source is considered negative.
Electromotive force (EMF) is a physical quantity that characterizes the action of external forces in current sources, this is the energy characteristic of a current source. For a closed loop, the EMF is defined as the ratio of the work of external forces to move a positive charge along a closed loop to this charge: 
In the International System, the unit of measurement for EMF is volt. With an open circuit, the EMF of the current source is equal to the electric voltage at its terminals.
Joule-Lenz law: the amount of heat released by a conductor with a current is determined by the product of the square of the current strength, the resistance of the conductor and the time of passage of the current through the conductor: 
When the electric field of the charge moves along the section of the circuit, it does work, which is determined by the product of the charge and the voltage at the ends of this section of the circuit:
DC power is a physical quantity that characterizes the speed at which the field performs work on the movement of charged particles along the conductor and is determined by the ratio of the work of the current during the time to this period of time: 
Kirchhoff rules, which are used to calculate branched DC circuits, the essence of which is to find, by the given resistances of the circuit sections and the EMF of the currents applied to them in each section.
The first rule is the rule of nodes: the algebraic sum of currents that converge at a node is a point at which there are more than two possible directions of current, it is zero
The second rule is the rule of circuits: in any closed circuit, in a branched electrical circuit, the algebraic sum of the products of the currents and the resistance of the corresponding sections of this circuit is determined by the algebraic sum of the EMF applied in it: 
A magnetic field- this is one of the forms of manifestation of an electromagnetic field, the specificity of which is that this field affects only moving particles and bodies that have an electric charge, as well as magnetized bodies, regardless of the state of their motion.
Magnetic induction vector is a vector quantity that characterizes the magnetic field at any point in space, which determines the ratio of the force acting from the magnetic field on the element of the conductor with an electric current to the product of the current strength and the length of the conductor element, which is equal in magnitude to the ratio of the magnetic flux through the cross-section of the area to the area of this cross-section.
In the International System of Units, the unit of induction is tesla (T).
Magnetic circuit is a collection of bodies or areas of space where the magnetic field is concentrated.
Magnetic flux (flux of magnetic induction) is a physical quantity that is determined by the product of the modulus of the magnetic induction vector by the area of a flat surface and by the cosine of the angle between the vectors of the normal to the flat surface / the angle between the normal vector and the direction of the induction vector.
In the International System of Units, the unit of magnetic flux is Weber (Wb).
Ostrogradsky-Gauss theorem for the flux of magnetic induction: the magnetic flux through an arbitrary closed surface is zero:
Ohm's law for a closed magnetic circuit: 
Magnetic permeability is a physical quantity that characterizes the magnetic properties of a substance, which is determined by the ratio of the modulus of the magnetic induction vector in the medium to the modulus of the induction vector at the same point in space in vacuum:
Magnetic field strength is a vector quantity that determines and characterizes the magnetic field and is equal to: 
Ampere force- This is the force that acts from the side of the magnetic field on the conductor with current. Ampere's elementary force is determined by the ratio:
Ampere's law: the modulus of the force acting on a small section of the conductor through which the current flows, from the side of a uniform magnetic field with induction making an angle with the element
Superposition principle: when at a given point in space, multiple sources form magnetic fields, the inductions of which are B1, B2, .., then the resulting field induction at this point is equal to: 
Gimbal Rule or Right Screw Rule: if the direction of the translational motion of the gimbal tip when screwing in coincides with the direction of the current in space, then the direction of the rotary movement of the gimbal at each point coincides with the direction of the magnetic induction vector.
Bio-Savart-Laplace law: determines the magnitude and direction of the magnetic induction vector at any point of the magnetic field created in vacuum by a conductor element of a certain length with a current: 
The movement of charged particles in electric and magnetic fields The Lorentz force is the force that affects a moving particle from the side of the magnetic field: 
Left hand rule:
- It is necessary to position the left hand so that the lines of magnetic induction enter the palm, and the extended four fingers are aligned with the current, then the thumb bent 90 ° will indicate the direction of the Ampere force.
- It is necessary to position the left hand so that the lines of magnetic induction enter the palm, and four outstretched fingers coincide with the direction of the particle speed with a positive charge of the particle or are directed in the direction opposite to the speed of the particle with a negative charge of the particle, then the thumb bent by 90 ° will show the direction the Lorentz force acting on a charged particle.
If there is a joint action on a moving charge of electric and magnetic fields, then the resulting force will be determined: 
Mass spectrographs and mass spectrometers are devices that are designed specifically for accurate measurements of the relative atomic masses of elements.
Faraday's law. Lenz's rule
Electromagnetic induction- this is a phenomenon, which consists in the fact that an EMF of induction occurs in a conducting circuit located in an alternating magnetic field.
Faraday's law: EMF of electromagnetic induction in the circuit is numerically equal and opposite in sign to the rate of change of the magnetic flux Ф through the surface bounded by this circuit: 
Induction current- This is the current that is formed if the charges under the action of the Lorentz forces begin to move.
Lenz's rule: the induction current that appears in a closed loop always has such a direction that the magnetic flux created by it through the area bounded by the loop tends to compensate for the change in the external magnetic field that caused this current.
The procedure for using the Lenz rule to determine the direction of the induction current:
Vortex field- this is a field in which the lines of tension are closed lines, the cause of which is the generation of an electric field by a magnetic one.
The work of a vortex electric field when a single positive charge moves along a closed fixed conductor is numerically equal to the EMF of induction in this conductor.
Toki Foucault- these are large induction currents that appear in massive conductors due to the fact that their resistance is low. The amount of heat that is released per unit time by eddy currents is directly proportional to the square of the frequency of the magnetic field change.
Self-induction. Inductance
Self-induction- This is a phenomenon that a changing magnetic field induces an EMF in the very conductor through which the current flows, which forms this field.
The magnetic flux Ф of the circuit with current I is determined by:
Ф = L, where L is the self-induction coefficient (current inductance).
Inductance is a physical quantity that is a characteristic of the EMF of self-induction that appears in the circuit when the current strength changes, it is determined by the ratio of the magnetic flux through the surface bounded by the conductor to the DC current in the circuit:
In the International System of Units, the unit for measuring inductance is henry (H).
EMF of self-induction is determined by: 
The energy of the magnetic field is determined:
The volumetric energy density of the magnetic field in an isotropic and non-ferromagnetic medium is determined by: 
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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 found 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 potential theory created by Green and Gauss. A lot of experimental research on electrostatics was carried out by Rees whose books were in the past the main guide in the study of these phenomena.
The dielectric constant
Finding the value of the dielectric coefficient K of any substance, a coefficient that is included in almost all formulas that have to be dealt with in electrostatics, can be done in very different ways. The most common methods are as follows.
1) Comparison of the capacitances of two capacitors that have the same size and shape, but in which one has an insulating layer of air, the other has a layer of the dielectric under test.
2) Comparison of the attraction between the surfaces of the capacitor, when a certain potential difference is imparted to these surfaces, but in one case there is air between them (attraction force = F 0), in the other case - the tested liquid insulator (attraction 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. Electric vibrations) propagating along the wires. According to Maxwell's theory, the speed of propagation of electric waves along the 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 in 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. The insulating medium is polarized along the induction tubes. Electrical displacements arise in it, which can be likened to the displacements of positive electricity in the direction of the axes of these tubes, and through each cross section of the tube passes an amount of electricity 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 (tension and pressure forces) that are in dielectrics when an electric field is excited in them. This question was first considered by Maxwell himself, and later by Helmholtz in more detail. Further development of the theory of this issue and the theory of electrostriction, which is closely connected with this (that is, the theory that considers phenomena that depend on the occurrence of special stresses 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.
Border conditions
Let us conclude our brief exposition of the most essential of the electrostriction department by considering the question of the refraction of induction tubes. Imagine in an electric field two dielectrics, separated from each other by some surface S, with dielectric coefficients K 1 and K 2.
Let at points Р 1 and Р 2, located infinitely close to the surface S on either side of it, the values of the potentials are expressed in terms of 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 the 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, there should be
K 1 d V 1 dn 1 + K 2 d V 2 dn 2 = 0. (31) (\ displaystyle K_ (1) (\ frac (dV_ (1)) (dn_ (1))) + K_ (2) ( \ frac (dV_ (2)) (dn_ (2))) = 0. \ qquad (31))We 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 beam entering from one medium into another. This consequence of the theory is justified by experience.
