... All predictions of electrostatics follow from its two laws.
But it is one thing to express these things mathematically, and quite another to
apply them with ease and with just the right amount of wit.
Richard Feynman
Electrostatics studies the interaction of stationary charges. Key experiments in electrostatics were carried out in the 17th and 18th centuries. With the discovery of electromagnetic phenomena and the revolution in technology that they produced, interest in electrostatics was lost for some time. However, modern scientific research shows the enormous importance of electrostatics for understanding many processes of living and inanimate nature.
Electrostatics and life
In 1953, American scientists S. Miller and G. Urey showed that one of the “building blocks of life” - amino acids - can be obtained by passing an electric discharge through a gas similar in composition to the primitive atmosphere of the Earth, consisting of methane, ammonia, hydrogen and vapor water. Over the next 50 years, other researchers repeated these experiments and obtained the same results. When short current pulses are passed through bacteria, pores appear in their shell (membrane), through which DNA fragments of other bacteria can pass in, triggering one of the mechanisms of evolution. Thus, the energy required for the origin of life on Earth and its evolution could indeed be the electrostatic energy of lightning discharges (Fig. 1).
How electrostatics cause lightning
At any given time, about 2,000 lightning flashes at different points on the Earth, 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 negative charge. Moreover, each of the discharges supplies the Earth with several tens of coulombs of electricity, and the amplitude of the current during a lightning strike ranges from 20 to 100 kiloamperes. High-speed photography showed that a lightning strike lasts only tenths of a second and that each lightning consists of several shorter ones.
Using measuring instruments installed on atmospheric probes, at the beginning of the 20th century, the Earth's electric field was measured, the strength of which at the surface turned out to be approximately 100 V/m, which corresponds to a total charge of the planet of about 400,000 C. The carrier of charges in the Earth's atmosphere are ions, the concentration of which increases with altitude and reaches a maximum at an altitude of 50 km, where under the influence of cosmic radiation an electrically conductive layer has formed - the ionosphere. Therefore, we can say that the Earth's electric field is the field of a spherical capacitor with an applied voltage of about 400 kV. Under the influence of this voltage, a current of 2–4 kA flows all the time 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 Earth's electrical capacitor is discharged, and during a thunderstorm it is charged.
A thundercloud is a huge amount of steam, some of which has condensed into tiny droplets or floes of ice. The top of a thundercloud can be at an altitude of 6–7 km, and the bottom can hang above the ground at an altitude of 0.5–1 km. Above 3–4 km, the clouds consist of ice floes of different sizes, since the temperature there is always below zero. These pieces of ice are in constant motion, caused by rising currents of warm air rising from below from the heated surface of the earth. Small pieces of ice are lighter than large ones, and they are carried away by rising air currents and collide with large ones along the way. With each such collision, electrification occurs, in which large pieces of ice are charged negatively, and small ones - positively. Over time, positively charged small pieces of ice collect mainly in the upper part of the cloud, and negatively charged large ones - at the bottom (Fig. 2). In other words, the top of the cloud is charged positively, and the bottom - negatively. In this case, positive charges are induced on the ground directly below the thundercloud. Now everything is ready for a lightning discharge, in which air breakdown occurs and the negative charge from the bottom of the thundercloud flows to the Earth.
It is typical that before a thunderstorm, the strength of the Earth's electric field can reach 100 kV/m, i.e., 1000 times higher than its value in good weather. As a result, the positive charge of each hair on the head of a person standing under a thundercloud increases by the same amount, and they, pushing away from each other, stand on end (Fig. 3).
Fulgurite - trace of lightning on the ground
During a lightning discharge, energy is released on the order of 10 9 –10 10 J. Most of this energy is spent on thunder, heating the air, 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 enough to cause a fire, kill a person or destroy a building. Lightning can heat the channel through which it moves to 30,000°C, which is much higher than the melting point of sand (1600–2000°C). Therefore, lightning, striking the sand, melts it, and the hot air and water vapor, expanding, form a tube from the molten sand, which after some time hardens. This is how fulgurites (thunder arrows, devil's fingers) are born - hollow cylinders made of melted sand (Fig. 4). The longest excavated fulgurites went underground to a depth of more than five meters.
How electrostatics protect against lightning
Fortunately, most lightning strikes occur between clouds and therefore do not pose a threat to human health. However, it is believed that lightning kills more than a thousand people around the world every year. At least in the USA, where such statistics are kept, about a thousand people suffer from lightning strikes every year and more than a hundred of them die. Scientists have long tried to protect people from this “punishment of God.” For example, the inventor of the first electric capacitor (Leyden jar), Pieter van Muschenbrouck, in an article on electricity written for the famous French Encyclopedia, defended traditional methods of preventing lightning - ringing bells and firing cannons, which he believed were quite effective.
In 1750, Franklin invented the lightning rod. In an attempt to protect the Maryland capitol building from a lightning strike, he attached a thick iron rod to the building, extending several meters above the dome and connected to the ground. The scientist refused to patent his invention, wanting it to begin serving people as soon as possible. The mechanism of action of a lightning rod is easy to explain if we remember that the electric field strength near the surface of a charged conductor increases with increasing curvature of this surface. Therefore, under a thundercloud near the tip of the lightning rod, the field strength will be so high that it will cause ionization of the surrounding air and a corona discharge in it. As a result, the likelihood of lightning hitting the lightning rod will increase significantly. Thus, knowledge of electrostatics not only made it possible to explain the origin of lightning, but also to find a way to protect against them.
The news of Franklin's lightning rod quickly spread throughout Europe, and he was elected to all academies, including the Russian one. However, in some countries the devout population greeted this invention with indignation. The very idea that a person could so easily and simply tame the main weapon of God’s wrath seemed blasphemous. Therefore, in different places people, for pious reasons, broke lightning rods.
A curious incident occurred in 1780 in a small town in northern France, where the townspeople demanded that the iron lightning rod mast be demolished and the matter came to trial. The young lawyer, who defended the lightning rod from the attacks of obscurantists, based his defense on the fact that both the human mind and his ability to conquer the forces of nature are of divine origin. Everything that helps save a life is for the good, the young lawyer argued. He won the case and gained great fame. The lawyer's name was... Maximilian Robespierre.
Well, now the portrait of the inventor of the lightning rod is the most coveted reproduction in the world, because it adorns the well-known hundred dollar bill.
Electrostatics that brings life back
The energy from the capacitor discharge not only led to the emergence of life on Earth, but can also restore life to people whose heart cells have stopped beating synchronously. Asynchronous (chaotic) contraction of heart cells is called fibrillation. Fibrillation of the heart can be stopped by passing a short pulse of current through all its cells. To do this, two electrodes are applied to the patient’s chest, through which a pulse is passed with a duration of about ten milliseconds and an amplitude of up to several tens of amperes. In this case, the discharge energy through 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 heart fibrillation is called a defibrillator. The simplest defibrillator is an oscillating circuit consisting of a capacitor with a capacity of 20 μF and a coil with an inductance of 0.4 H. By charging the capacitor to a voltage of 1–6 kV and discharging it through the coil and the patient, whose resistance is about 50 ohms, you can obtain the current pulse necessary to bring the patient back to life.
Electrostatics giving light
A fluorescent lamp can serve as a convenient indicator of electric field strength. To verify this, while in a dark room, rub the lamp with a towel or scarf - as a result, the outer surface of the lamp glass will be charged positively, and the fabric - negatively. As soon as this happens, we will see flashes of light appearing in those places of the lamp that we touch with a charged cloth. Measurements have shown that the electric field strength inside a working fluorescent lamp is about 10 V/m. At this intensity, free electrons have the necessary energy to ionize mercury atoms inside a fluorescent lamp.
The electric field under high-voltage power lines - power lines - can reach very high values. Therefore, if at night a fluorescent lamp is stuck into the ground under a power line, it will light up, and quite brightly (Fig. 5). So, using the energy of an electrostatic field, you can illuminate the space under power lines.
How electrostatics warn of fire and make smoke cleaner
In most cases, when choosing the type of fire alarm detector, preference is given to a smoke 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 the building about the danger. Smoke detectors use ionization or photoelectric principle to detect smoke in the air.
Ionization smoke detectors contain an α-radiation source (usually americium-241) that ionizes air between metal electrode plates, 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 smoke microparticles that appear there bind to the ions, neutralize their charge and thus increase the resistance between the electrodes, to which the electrical circuit reacts by sounding an alarm. Sensors based on this principle demonstrate very impressive sensitivity, reacting even before the very first sign of smoke is detected by a living creature. It should be noted that the radiation source used in the sensor does not pose any danger to humans, since alpha rays cannot pass even through a sheet of paper and are completely absorbed by a layer of air several centimeters thick.
The ability of dust particles to electrify is widely used in industrial electrostatic dust collectors. A gas containing, for example, soot particles, rising upward, passes through a negatively charged metal mesh, as a result of which these particles acquire a negative charge. Continuing to rise upward, the particles find themselves in the electric field of positively charged plates, to which they are attracted, after which the particles fall into special containers, from where they are periodically removed.
Bioelectrostatics
One of the causes of asthma is the waste products of dust mites (Fig. 6) - insects about 0.5 mm in size that live in our house. Research has shown that asthma attacks are caused by one of the proteins that these insects secrete. The structure of this protein resembles a horseshoe, both ends of which are positively charged. The electrostatic repulsive forces between the ends of such a horseshoe-shaped protein make its structure stable. However, the properties of a protein can be changed by neutralizing its positive charges. This can be done by increasing the concentration of negative ions in the air using any ionizer, for example a Chizhevsky chandelier (Fig. 7). At the same time, the frequency of asthma attacks decreases.
Electrostatics helps not only to neutralize proteins secreted by insects, but also to catch them themselves. It has already been said that the hair “stands on end” if it is charged. You can imagine what insects experience when they find themselves electrically charged. The thinnest hairs on their legs diverge in different directions, and the insects lose the ability to move. The cockroach trap shown in Figure 8 is based on this principle. Cockroaches are attracted to sweet powder that is previously electrostatically charged. Powder (it is white in the picture) is used to cover the inclined surface around the trap. Once on the powder, the insects become charged and roll into the trap.
What are antistatic agents?
Clothes, carpets, bedspreads, etc. objects are charged after contact with other objects, and sometimes simply with jets of air. In everyday life and at work, charges generated in this way are often called static electricity.
Under normal atmospheric conditions, natural fibers (cotton, wool, silk and viscose) absorb moisture well (hydrophilic) and therefore slightly conduct electricity. When such fibers touch or rub against other materials, excess electrical charges appear on their surfaces, but for a very short time, as the charges immediately flow back through the wet fibers of the fabric containing various ions.
Unlike natural fibers, synthetic fibers (polyester, acrylic, polypropylene) do not absorb moisture well (hydrophobic), and there are fewer mobile ions on their surfaces. 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, so when it comes into contact, for example, with a comb, it becomes charged with electricity and begins to repel each other.
To get rid of static electricity, the surface of clothing or other items can be lubricated with a substance that retains moisture and thereby increases the concentration of mobile ions on the surface. After such treatment, the resulting electrical charge will quickly disappear from the surface of the object or be distributed over it. The hydrophilicity of a surface can be increased by lubricating it with surfactants, whose molecules are similar to soap molecules - one part of a very long molecule is charged, and the other is not. Substances that prevent the appearance of static electricity are called antistatic agents. For example, ordinary coal dust or soot is an antistatic agent, therefore, in order to get rid of static electricity, so-called lamp black is included in the impregnation of carpeting and upholstery materials. For the same purposes, up to 3% natural fibers and sometimes thin metal threads are added to such materials.
Electrodynamics, as a serious and diverse branch of modern physics, is divided into several main areas. Electrodynamics is designed to study the concept of electric charge. Electric charge is tightly coupled to the electromagnetic field. It is its material source of origin. The electromagnetic field itself is an internal characteristic of elementary particles that are in constant interaction with each other, which gives rise to various physical phenomena and properties of bodies. Electric charge is a scalar physical quantity and determines electromagnetic interaction.
Figure 1. Concept of electrostatics. Author24 - online exchange of student works
According to the first model of particle interaction, any charged particle is capable of exciting the surrounding space around it. In this case, any other particle that finds itself in such a disturbed space will experience a certain force. In this case, it is customary to consider a particle caught in an electromagnetic field. The fact of the presence of a charged particle must necessarily be associated with the source of this force. This is the electrical component of the process. The magnetic base will be associated with its movement. Each charged body can be considered as a collection of charged particles that can create an electromagnetic field.
Electrostatics – section of electrodynamics
Electrostatics, as a branch of electrodynamics, considers the interaction of stationary electric charges passed through an electrostatic field. Charges are stationary relative to another frame of reference, so all conclusions can be drawn at an approximate level, but it always moves with some speed relative to another frame of reference.
In total, it is customary to distinguish between two types of electric charges:
- positive;
- negative.
Elementary particles can serve as carriers of such electric charges. Their composition must certainly include atoms. All atoms consist of:
- negative charge (electron);
- positive charge (proton).
They have some characteristic features. The unit of measurement for charge is the coulomb. A body is charged if it contains different numbers of positive and negative elementary particles.
For the manifestation of an electromagnetic field, the action of electromagnetic forces is necessary. It consists in the formation:
- friction forces;
- elastic forces;
- action of electromagnetic forces at the level of elementary particles.
When studying the basics of electrostatics, it is impossible not to dwell on the concept of electrification of bodies. This is a method of producing charged particles by contact. In this case, the bodies will be mutually charged, but they will become equal in magnitude and opposite in sign of charge.
Basic concepts of electrostatics
The basic law of electrostatics is Coulomb's law. It is defined as the force of interaction between two stationary point charged bodies. It occurs under vacuum conditions and is directly proportional to the product of the charge modules, and also inversely proportional to the square of the distance between them.
Bodies are considered point bodies at the moment when the distance between them is much greater than the size of the bodies themselves. Bodies interact according to Coulomb's law if they have electrical charges.
Electric field strength is a certain quantitative characteristic of the electric field. It combines the characteristics of a relationship of power. With this parameter, the field acts on a point charge. It is correlated with the magnitude of a given charge. Also, the field strength cannot depend on the amount of charge introduced. It only characterizes the entire electric field in general. The direction of the tension vector must completely coincide with the direction of the force vector that acts on the positive charge, and it is also opposite to the direction of the force that acts on the negative charge.
Power lines
To formulate the concept of an electric field at a theoretical level, they use lines of force. Similar lines are drawn so that the direction of the tension vector at each point coincides with the direction of the tangent to the line of force. Lines of force may have a number of characteristic features and properties.
They cannot intersect in an electrostatic field. These lines turn out to be directed towards negative charges from positive charges. When depicting electric field lines, they resort to different thicknesses of application. They must be proportional to the magnitude of the field strength vector. Their density increases according to tension and is always proportional to it.
At a certain point in space, it is customary to draw only one line of force. This is due to the fact that the electric field strength at this point can only be specified unambiguously.
Figure 2. Concept of electrodynamics. Author24 - online exchange of student works
If the electric field is uniform, then the intensity vector is also at the same level as it. This manifests itself at all points of the field in space. Such a field is created by a flat plate capacitor. They must be charged with the same amount of charge, separated by a layer of dielectric, but this distance must be created less than the size of the plates themselves.
Electrical capacity characterizes the ability of conductors to accumulate electrical charge at a certain point. It depends on the shape, the relative arrangement of charges, the size of the conductors, as well as the characteristic properties of the medium between the conductors.
The basic formulas of electrostatics are as follows. Here are presented the equations for the interaction of charges, electric potential, work of the electrostatic field, electric capacitance, as well as electric field strength.
Figure 3. Basic formulas in electrostatics. Author24 - online exchange of student works
Electrodynamics also studies the lines of force of the electrostatic field, the operation of the electrostatic field and equipotential surfaces. The basics of an electrical circuit, the laws of direct current, resistance and other definitions characteristic of this branch of physics are also introduced.
Federal Agency for Education State Educational Institution of Higher Professional Education Tula State Pedagogical University
named after L. N. Tolstoy
Yu. V. Bobylev V. A. Panin R. V. Romanov
GENERAL PHYSICS COURSE
electrodynamics
Short course of lectures
Approved by the Educational and Methodological Association
in areas of teacher education of the Ministry of Education and Science of the Russian Federation as a teaching aid
for students of higher educational institutions studying in the direction 540200 (050200)
"Physics and mathematics education"
Tula Publishing House TSPU im. L. N. Tolstoy
BBK 22.3ya73 B72
Reviewer –
Professor Yu. F. Golovnev (Tashkent State Pedagogical University named after L. N. Tolstoy)
Bobylev, Yu. V.
B72 General physics course. Electrodynamics: A short course of lectures / Yu. V. Bobylev, V. A. Panin, R. V. Romanov. – Tula: Tula Publishing House. state ped. unta im. L. N. Tolstoy, 2007.– 107 p.
This textbook is a short lecture course on electromagnetism and contains the necessary material that fully complies with the State educational standard.
The manual is intended mainly for students who, for one reason or another, cannot attend or attend classroom classes irregularly and are engaged in self-education, including distance learning.
By reducing the mathematical part, the manual can be positioned for students of non-physical specialties.
© Yu. V. Bobylev, V. A. Panin, R. V. Romanov,
© Publishing house TSPU im. L. N. Tolstoy,
Preface........................................................ ........................................... |
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Introduction........................................................ ................................................... |
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Lecture 1. Electric charge.................................................... .............. |
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Lecture 2. Coulomb's Law.................................................... ........................... |
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Lecture 4. Gauss's theorem.................................................... ........................ |
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Lecture 5. Electric field potential.................................................. |
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Lecture 6. Electric field potential (continued)................... |
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Lecture 7. Conductors in an electric field................................................. |
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Lecture 8. Dielectrics in an electric field................................................. |
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Lecture 9. Electric capacitance. Capacitors........................ |
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Lecture 10. Electrostatic energy.................................................... |
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Lecture 11. Direct current. Basic concepts and laws.. ............ |
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Lecture 12. Electric circuits.................................................... .............. |
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Lecture 13 Current in metals.................................................... ........................ |
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Lecture 14. Current in a vacuum.................................................... ........................ |
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Lecture 15. Current in gases. ........................................................ ........................ |
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Lecture 16. Current in electrolytes. ........................................................ ......... |
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Lecture 17. Basic laws of magnetism. ......................................... |
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Lecture 18. Basic laws of magnetism (continued)................ |
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Lecture 19. Movement of charged particles in a magnetic field.......... |
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Lecture 20 Electromagnetic induction. ............................................ |
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Lecture 21. Electric oscillatory circuit.................................... |
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Lecture 22. Alternating current.................................................... .................... |
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Lecture 23. Electric field.................................................... .............. |
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Lecture 24. Maxwell's equations.................................................... .......... |
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Lecture 25. Electromagnetic waves.................................................... .... |
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Conclusion................................................. ............................................ |
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Literature................................................. ........................................... |
Preface
The authors of this manual work at the Faculty of Mathematics, Physics and Computer Science of Tula State Pedagogical University. L.N. Tolstoy and have already repeatedly taught various disciplines and special courses related to electromagnetic processes, including phenomena in nonequilibrium material media, as part of courses in general and theoretical physics.
Teaching experience, formed by significant work experience (from 20 to 25 years), suggested the concept of creating a single end-to-end course in electrodynamics. It should include, without duplication or repetition, which is quite important, all the topics studied in the courses of general and theoretical physics, such as “Electricity and Magnetism”, “Electrodynamics and the Fundamentals of SRT”, “Electrodynamics of Continuum Media” and so on.
Such a course will allow you to maintain a unified style of presentation and design, the same notation, a unified system of units, and a similar use of mathematical apparatus, which will certainly simplify the perception of this difficult material by students.
It should be noted that the scientific interests of the authors lie in the areas of electrodynamics of highly nonequilibrium plasma, nonlinear phenomena in electrodynamic systems and structures of various natures, certain issues of plasma electronics and radiophysics, which, of course, makes this manual as close as possible to modern scientific achievements.
The implementation of this concept began in 2002 with the release of a textbook for the course “Electricity and magnetism: a course of lectures. Part 1. Electrostatics", which was approved by the Ministry of Education as a teaching aid for students of physics and mathematics.
Teaching using this manual has shown its undoubted effectiveness and demand among students. In 2004, a collection of problems for the course “Electricity and Magnetism” was published. The preparation of these materials in WEB document format made it possible to use them not only for full-time students, but also for distance learning.
In this manual, a more concise “telegraphic” style of presentation is used, and the language, generally speaking, is far from academic and is as close as possible to colloquial, as, in fact, it should be, since the material is a record of what the student heard and saw at the lecture.
A large number of drawings are used, which, however, are schematic and simplified. Some complex formulas are given with detailed conclusions, which will be especially valuable for students graduating from rural schools. In addition, according to the authors, the manual contains a significant number of examples of problem solutions that make it easier to understand
theoretical material and contributing to the development of practical skills of the future teacher.
IN The International System of Units (SI) is used as the main one.
IN In general, the material corresponds to the minimum specified in the State Educational Standard for Higher Professional Education and the curriculum.
The authors believe that this textbook on electromagnetism will help students who, for one reason or another (we will consider it valid) cannot attend or attend classroom classes irregularly and are engaged in self-education. There are more and more such students, but getting them to read traditional textbooks and scrupulously select the necessary information from them, taking into account the realities of the present time, is very problematic. This manual contains the necessary already selected material that fully complies with the State Educational Standard, so that the average student receives a positive mark on the exam without the use of additional literature.
For students who want to gain deeper knowledge and plan to continue their studies in a master's program, at the end of this manual there is a fairly comprehensive list of useful literature.
You should not think that this manual is only suitable for lagging students. It is intended for all students, the only difference being that a student who attended the lecture and a student who missed the lecture will have to work with this manual in different ways.
Moreover, in the context of the transition to two-level education and in the conditions of increasing penetration and implementation of the basic ideas of the Bologna process, such manuals, which on the one hand are sufficiently unified to meet the strict requirements of the state standard, and on the other hand, have an undoubted “stamp” of individuality and creative views authors will be more and more in demand on the “student market”.
It should also be noted that this manual, while shortening the mathematical part, can be positioned for students of non-physical specialties.
Tula, April 2007
Introduction
1. Electrodynamics as a science
Definition: Electrodynamics– a science that studies the behavior of the electromagnetic field that interacts between electric charges.
2. Historical background
Here you can cite almost the entire course on the history of physics, to which we refer you.
3. Theory of long- and short-range action
For a long time, physics was dominated by the theory of long-range action, which, based on mathematical laws, described the interaction of bodies without indicating the mechanism of this interaction. This is due to the fact that Newton’s well-formulated laws perfectly described all mechanical phenomena, without themselves being subject to any explanation. The mechanical approach extended to other branches of physics (Coulomb's law). The works of Ostrogradsky, Gauss, Laplace, etc. this theory acquired a complete mathematical form. At the same time, scientists were concerned about the question of how and with what help the interaction is transmitted. Faraday introduced the concept of a field, which is the carrier of interaction. For a long time, theories existed equally.
In quasi-static fields they lead to the same results. And only after the experiments of Hertz and Popov with rapidly varying fields was the question clearly resolved in favor of the theory of short-range action. It is believed that interactions between charges are carried out using an electromagnetic field that propagates in space. In a vacuum the field propagates at a speed
c=299792458 m/s≈3.00·108 m/s.
Electric charge
1. General concepts
Definition: Electric charge is a physical quantity that determines the electromagnetic field through which the interaction between charges occurs.
Despite the various ways to obtain a charge, there are only two types of electricity: “glass” and “resin” (“+” and “–”). Although there is an opinion that in fact this is an excess or lack of electricity of one kind, namely negative. In nature, the amount of positive electricity is approximately equal to the amount of negative electricity.
2. Methods for obtaining electrified bodies
3. Charge measurement
Definition: A test charge is a charge that does not introduce distortion into the existing field.
Let there be some electric field. We place a test charge at some point in the field. The field will act on it with some force.
We introduce another test charge into this field. If the forces are directed in one direction, then the charges are of the same name; if not, then they are opposite.
F 1 = F 2 q 1 q 2
F 1 = const = q 1 F 2 q 2
Knowing the ratio of forces, we also know the ratio of charges, and, taking one of the charges as a standard, we indicate the fundamental method for measuring charges.
4. Charge unit
Definition: 1 Coulomb is an SI unit of electric charge equal to the charge flowing through the cross-section of a conductor in 1 s at a constant current of 1 A.
5. Law of conservation of charge
If an energetic photon falls on a closed system, a paired electric charge can be created. In total, the charge of the system will not change. All experiments show that charge has the inherent property of being conserved, so this position is elevated to the rank of a postulate.
Law: In a closed system, electric charge is a constant quantity.
∑ qi = const.
i= 1
6. Charge the Earth
The Earth's charge is negative.
q = − 6 105 C .
7. Charge invariance
Basically, charges are measured by comparing forces. Force is an invariant, i.e. it is the same in different reference systems. Therefore, the charge ratio is also invariant. And if the charge standard is the same, then we can say that the charge has the same quantitative value in different reference systems.
8. Charge discreteness
Any charge can be represented in the form
q = N e , N = 0, ± 1, ± 2, ...
|e| = 1.6021892(46)·10-19 C - elementary charge
Electric charge is said to be discrete or quantized, i.e. There is a certain minimum portion of charge that cannot be divided further.
9. Models of charged bodies
As a rule, it is believed that the charge is continuously “smeared” over the body and the concepts of physically infinitesimal charge and volume are introduced.
<< dV
< |
10− 27 |
÷ 10 |
− 30 m 3 ; |
<< dq << Q ; |
||||||||||||
Bulk Density |
Superficial |
Linear density |
||||||||||||||
density |
||||||||||||||||
ρ = |
= ρ(x, y, z) |
σ = dq |
τ = dq |
|||||||||||||
Q = ∫ ρ (x, y, z) dV |
Q = ∫ σ dS |
Q = ∫ τ dl |
||||||||||||||
V body |
S body |
L body |
||||||||||||||
10. Point charge
Definition: Point charge is called a material point that has a charge.
The point charge density can be written as a formula;
ρ (r) = q δ (r − r 0 ).
Here r 0 is the radius vector that determines the position of the point charge; δ (r − r 0 )
– Dirac delta function.
11. Delta function or Dirac function.
In the one-dimensional case, this function is defined as follows:
0, x ≠ 0 |
||
∫ δ (x) dx = 1 |
||
δ(x) = ∞, x = 0 |
||
It also follows that
Electrostatics is a branch of the science of electricity that studies stationary electric charges. It is based on 3 main facts: the existence of two types of charges, the presence of interaction between them and the principle of superposition (the interaction of two charges is not affected by the third).
And so in nature there are two types of electric charges. Conventionally, one of them is assigned a plus sign “+” and the other, respectively, a minus sign “-”. There is an electric field around these charges, and if these charges are stationary, then the field is called electrostatic.
Figure 1 Negative and positive charges.
Electric charge is a discrete quantity. That is, it consists of elementary charges of a certain size. And the total charge of any body is a multiple of this elementary charge.
When studying charges in electrostatics, averaging methods are used, both in time and in space. This allows us to consider charges in chaotic thermal motion as stationary.
All charges, both positive and negative, are part of the molecules of a substance. Thus, any body has a large number of charges. But the phenomena of interaction of electrostatic charges can be observed only if the body has an excess (deficiency) of charges of the same sign.
The law of conservation of charge states that if a system is closed, then the total charge in it is unchanged. These charges can be distributed in any way within the system, which will not affect the charge of the system as a whole.
The unit of measurement for the field created by electric charges is intensity. It is depicted graphically in the form of lines of force. The density of the field lines indicates the magnitude of the field strength.
Figure 2 field between unlike charges.
Like charges always repel, and unlike charges attract. Between charges of sizes that can be neglected (point charges), the so-called Coulomb force acts. Coulomb's law determines the force of interaction between two electric charges, depending on their magnitude and the distance between them.
Formula 1 Coulomb's law
The electrostatic field is potential. This means that the work done to move a charge from one point to another does not depend on the shape of the charge path. If one of the points is at infinity, then the concept of electric potential can be introduced. It determines the work spent on moving a charge from infinity to a given point in space.
And finally, let's talk about the principle of field superposition. The essence of the principle is that the resulting field of several point charges will be the vector sum of the fields of each of the charges separately. That is, the field of the third charge does not affect the fields of the other two charges.
Figure 3 principle of field superposition
The main problems that electrostatics solves are determining the charge distribution over a surface, knowing the potential of the surface or its total charge. Finding the energy of a system of conductors, knowing their charges and potentials. And also the study of the behavior of various substances in an electric field.
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The foundation of electrostatics was laid by the work of Coulomb (although ten years before him, the same results, even with even greater accuracy, were obtained by Cavendish. The results of Cavendish’s work were kept in the family archive and were published only a hundred years later); the law of electrical interactions discovered by the latter made it possible for Green, Gauss and Poisson to create a mathematically elegant theory. The most essential part of electrostatics is the theory of potential, created by Green and Gauss. A lot of experimental research on electrostatics was carried out by Rees, whose books in the past constituted the main guide for the study of these phenomena.
The dielectric constant
Finding the value of the dielectric coefficient K of any substance, a coefficient included in almost all formulas that one has to deal with in electrostatics, can be done in very different ways. The most commonly used methods are the following.
1) Comparison of the electrical capacitances of two capacitors having the same size and shape, but in one of which the insulating layer is a layer of air, in the other - a layer of the dielectric being tested.
2) Comparison of attractions between the surfaces of a capacitor, when a certain potential difference is imparted to these surfaces, but in one case there is air between them (attractive force = F 0), in the other case, the test liquid insulator (attractive force = F). The dielectric coefficient is found by the formula:
K = F 0 F .(\displaystyle K=(\frac (F_(0))(F)).)
3) Observations of electric waves (see Electrical oscillations) propagating along wires. According to Maxwell's theory, the speed of propagation of electric waves along wires is expressed by the formulaV = 1 K μ .
(\displaystyle V=(\frac (1)(\sqrt (K\mu ))).)Usually, the lengths of standing electric waves that arise in parts of the same wire located in the air and in the test dielectric (liquid) are compared. Having determined these lengths λ 0 and λ, we obtain K = λ 0 2 / λ 2. According to Maxwell’s theory, it follows that when an electric field is excited in any insulating substance, special deformations occur inside this substance. Along the induction tubes, the insulating medium is polarized. Electrical displacements arise in it, which can be likened to the movements of positive electricity 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 (forces of tension and pressure) that appear in dielectrics when an electric field is excited in them. This question was first considered by Maxwell himself, and later in more detail by Helmholtz. Further development of the theory of this issue and the closely connected theory of electrostriction (that is, the theory that considers phenomena that depend on the occurrence of special voltages in dielectrics when an electric field is excited in them) belongs to the works of Lorberg, Kirchhoff, P. Duhem, N. N. Schiller and some others
Border conditions
Let us complete our brief presentation of the most significant aspects of electrostriction by considering the issue of refraction of induction tubes. Let us imagine two dielectrics in an electric field, separated from each other by some surface S, with dielectric coefficients K 1 and K 2.
Let at points P 1 and P 2 located infinitely close to the surface S on either side of it, the magnitudes of the potentials are expressed through V 1 and V 2 , and the magnitudes of the forces experienced by a unit of positive electricity placed at these points through F 1 and F 2. Then for a point P lying on the surface S itself, there must be V 1 = V 2,d V 1 d s = d V 2 d s , (30) (\displaystyle (\frac (dV_(1))(ds))=(\frac (dV_(2))(ds)),\qquad (30))
if ds represents an infinitesimal displacement along the line of intersection of the tangent plane to the surface S at point P with the plane passing through the normal to the surface at this point and through the direction of the electric force in it. On the other hand, it should beLet us denote by ε 2 the angle made by the force F2 with the normal n2 (inside the second dielectric), and by ε 1 the angle made by the force F 1 with the same normal n 2 Then, using formulas (31) and (30), we find
t g ε 1 t g ε 2 = K 1 K 2 .(\displaystyle (\frac (\mathrm (tg) (\varepsilon _(1)))(\mathrm (tg) (\varepsilon _(2))))=(\frac (K_(1))(K_( 2))).)
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