Electrostatics

The Ministry of Science and Higher Education of the Russian Federation 
Kazan National Research Technological University 
V. Arkhipov
ELECTROSTATICS
Tutorial 
Kazan 
KNRTU Press 
2024 

UDC 537.2(075) 
Published by the decision of the Editorial Review Board  
of the Kazan National Research Technological University 
Reviewers: 
Doctor of Engineering Sciences, Professor A. Turanov 
Ph.D. in Physics and Mathematics, Associate Professor I. Lunev 
Arkhipov V. 
Electrostatics : Tutorial / V. Arkhipov; The Ministry of Education and 
Science of the Russian Federation, Kazan National Research Technological 
University. – Kazan : KNRTU Press, 2024. – 88 p. 
ISBN 978-5-7882-3492-2 
The tutorial outlines the main issues of the General Physics course (sections Electricity and Magnetism). It contains theoretical material on the topics: electric field, conductors and dielectrics in an electric field, electric field energy, analysis of solutions 
to typical problems, and self-check questions.  
The tutorial is intended for students studying Bachelor program 18.03.01 Chemical 
Technology of all specialties of mechanical and techno-logical profiles in order 
to strengthen the theoretical foundations of the General Physics course.  
The tutorial is prepared by the Department of Physics. 
537.2(075) 
ISBN 978-5-7882-3492-2 
© V. Arkhipov, 2024 
© Kazan National Research Technological 
University, 2024 
2

C O N T E N T S
Introduction ........................................................................................................... 5 
1. ELECTRIC FIELD ........................................................................................... 9 
1.1. Electric charge 
...................................................................................... 9 
1.2. Coulomb's law .................................................................................... 11 
1.3. Electrostatic field strength 
.................................................................. 12 
1.4. Electrostatic field potential ................................................................ 14 
1.5. Theorem on the circulation of the field strength vector 
..................... 16 
1.6. Relationship between electric field strength and field potential 
........ 17 
1.7. Principle of electric field superposition ............................................. 18 
1.8. Graphic representation  of electrostatic fields 
.................................... 20 
1.9. Ostrogradsky–Gauss theorem ............................................................ 22 
1.10. Field of an infinite homogeneously charged plane. ......................... 24 
1.11. Field of an infinite uniformly charged thread .................................. 26 
1.12. Problems 
........................................................................................... 27 
Self-check questions 
.................................................................................. 34 
2. CONDUCTORS  IN AN ELECTROSTATIC FIELD ................................... 35 
2.1. Properties of electron gas in metals ................................................... 35 
2.2. Electric field inside and near the surface of the conductor 
................ 36 
2.3. Van de Graaff electrostatic generator ................................................ 39 
2.4. Electrical induction and induced charges 
........................................... 41 
2.5. Electrostatic protection 
....................................................................... 41 
2.6. Capacitance ........................................................................................ 43 
2.7. Problems 
............................................................................................. 46 
Self-check questions 
.................................................................................. 49 
3. ELECTRIC FIELD IN DIELECTRICS 
.......................................................... 51 
3.1. Electric dipole .................................................................................... 51 
3.2. Dipole in a uniform electrostatic field ............................................... 53 
3.3. Polarization of dielectrics 
................................................................... 55 
3.4. Mechanisms of polarization of dielectrics ......................................... 57 
3.5. The relationship between dielectric susceptibility and permeability 
. 60 
3.6. Electric displacement vector or electric induction vector 
.................. 61 
3.7. Conditions on the interface between two dielectrics ......................... 62 
3.8. Ferroelectrics or segnetoelectrics 
....................................................... 65 
3.9. Electrets .............................................................................................. 67 
3.10. Problems 
........................................................................................... 68 
Self-check questions 
.................................................................................. 74 
3 

4. ELECTRIC FIELD ENERGY ........................................................................ 75 
4.1. Energy of a charged conductor .......................................................... 75 
4.2. Energy of a system of fixed charged conductors ............................... 76 
4.3. Energy of a dipole in an electric field ................................................ 77 
4.4. Energy of a charged capacitor 
............................................................ 78 
4.5. Volumetric density of electric field energy 
........................................ 79 
4.6. Problems 
............................................................................................. 80 
Self-check questions 
.................................................................................. 84 
Conclusion 
........................................................................................................... 85 
Bibliography 
........................................................................................................ 86 
4 

Knowledge is a map that guides us, while 
imagination is the territory where we can 
roam freely and search for answers and opportunities. Imagination knows no restraint, 
and it is the power that puts knowledge to use 
Albert Einstein 
I N T R O D U C T I O N
The first information about electrical phenomena was already known 
in ancient times. It was known that amber, if rubbed with wool, attracts light 
objects. The word “electricity” itself comes from the Greek word “electron”, 
which means amber. William Gilbert1 established that many bodies, just like 
amber, after rubbing, could attract small objects and called such phenomena 
electrical (from the Latin electricus, “amber”).  
The first electric machine was invented in 1663 by the German scientist Guericke2. Guericke’s device was a ball of sulfur. The rotating ball was 
pressed by hand, and it became electrified by friction. In 1729, the English 
physicist Stephen Gray3 conducted the first experiments on the transmission 
of electricity, discovered the phenomenon of electrical conductivity, and separated substances into conductors and non-conductors (insulators).  
1 William Gilbert (1544–1603) was an English physicist and physician. He studied magnetic and electrical phenomena and was the first to coin the term “electric.” In 1600, he 
wrote the book “On the Magnet, Magnetic Bodies, and the Great Magnet—the Earth,” 
where he first consistently examined magnetic and many electrical phenomena. 
2 Otto von Guericke (1602–1686) was a German physicist, engineer, and philosopher. 
From 1646 – burgomaster of Magdeburg. In 1654, he conducted a famous experiment 
with the Magdeburg hemispheres, proving the presence of air pressure. In 1663, he invented one of the first electrostatic generators that produced electricity by friction—
a ball of sulfur rubbed by hand. 
3 Stephen Gray (1666–1736) was an English physicist. Basic research in the field of 
electricity. He discovered the phenomenon of electrical conductivity in 1729, establishing that electricity can be transmitted from one body to another through a metal wire or 
spinning thread. 
5 

In 1745, the first electrical capacitor, the Leyden jar, was invented by 
Pieter van Musschenbrouck1 and Kleist. The Leyden jar made it possible to 
accumulate and store relatively large electrical charges. With the help of 
the Leyden jar, it was possible for the first time to artificially obtain an electric spark. Intensive research into electrical phenomena begins. The effect of 
electric discharge on the human body is being studied. Works appear on the 
“electricity of the human body” and “electrical healing energy.” 
Work is underway to study atmospheric electricity. Franklin2 proves 
the identity of terrestrial and atmospheric electricity, proves the electrical nature of lightning, and invents a lightning rod. He developed a “unitary theory” of electrical phenomena, according to which electricity is a special thin 
liquid that permeates all bodies. If there is an excess of electrical fluid in 
the body, then it has a positive charge; if there is a deficiency, then the charge 
of the body will be negative. Richmann3 and Lomonosov4 develop the theory 
of atmospheric electricity. 
The law of conservation of electric charge is formulated by St. Petersburg academician Franz Epinus5: “You cannot create one type of electricity 
without creating another. As soon as positive electricity arises, negative electricity arises simultaneously with it, and one cannot be obtained without 
the other.” Epinus suggests that the forces of interaction between electric 
1 Pieter van Musschenbroek (1692‒1761) was a Dutch physicist, creator of the “Leyden 
jar.”  Musschenbroek drew attention to the physiological effect of current by conducting 
experiments with a Leyden jar. 
2 Benjamin Franklin (1706‒1790) was an American physicist, political, and public figure. Invented (1750) a lightning rod, proved (1753) the electrical nature of lightning 
(experiment with a kite). Introduced the concepts of positive and negative electricity 
(charge), and introduces their designations “+” and “−”. 
3 Georg Wilhelm Richmann (1711–1753) was a Russian physicist. Invented the first 
electrical measuring device in 1745 – the electric pointer. In 1752–53, he investigated 
atmospheric electricity and built a “thunder machine” at his home. He died during an 
experiment from a ball lightning strike. 
4 Mikhail Vasilievich Lomonosov (1711–1765) was an outstanding Russian scientist. 
His works are devoted to physics, chemistry, astronomy, mining, metallurgy, etc. Together with G.W. Richmann, he conducted research in the field of electricity. He developed a theory of the formation of atmospheric electricity, the origin of which he associated with ascending and descending air currents. 
5 Franz Epinus (1724–1802) was a physicist, studied electrical and magnetic phenomena. He expressed the idea (1759) that electric and magnetic forces are inversely proportional to the square of the distance. 
6 

charges, like the forces of gravity, should be inversely proportional to 
the squares of the distances between them. 
The beginning of the scientific approach to the study of elec-tromagnetic phenomena was laid by Charles Coulomb1, who established in 1785 
the law of interaction of electric charges.  
In the works of Oersted2 and Ampere3 (1820), the laws of magnetic 
interaction of conductors with current were investigated and established, and 
the theory of magnetism was developed.  
In 1831, Faraday4 discovered the phenomenon of electromagnetic induction, a phenomenon that forms the basis of modern electrical power engineering. In the 19th century, numerous physicists tried to explain the phenomena of electricity and magnetism on the basis of classical mechanics. 
In 1870, James C. Maxwell5 formulated his theory of electrodynamics, which 
is based on the idea of fields and waves, to deal with these phenomena. 
The laws of electromagnetic phenomena are generalized in the system of 
Maxwell’s equations (1873). Maxwell’s equations for electromagnetism 
have been called the “second great unification in physics”, where the first 
one was realized by Isaac Newton. 
                                                          
 
1 Charles Coulomb (1736–1806) was a French physicist and military engineer. He graduated (1761) from the school of military engineers and was in military service all the time. 
By studying the torsion of silk and metal threads, he established the laws of elastic torsion. 
He established that the elastic force depends on the material of the thread, is proportional 
to the angle of twist, is the fourth power of the diameter of the thread, and is inversely 
proportional to its length. In 1784, he built a very sensitive device for measuring force—
a torsion balance. Using torsion balances, he studied the interaction of point charges and 
established in 1785 the fundamental law of electrostatics—Coulomb's law. 
2 Hans Christian Oersted (1777–1851) was a Danish physicist and chemist who discovered that electric currents create magnetic fields, which was the first connection found 
between electricity and magnetism. 
3 Andre-Marie Ampere (1775–1836) was a French physicist, mathematician, and natural 
scientist. Ampere created the first theory that expressed the connection between electrical and magnetic phenomena, introduced the concept of electric current into physics, 
and insightfully suggested that magnetism was caused by electric currents “at the molecular level.” James Maxwell called Ampere the "Newton of electricity." 
4 Michael Faraday (1791–1867) was an English physicist. Research in electricity, magnetism, magnetooptics, and electrochemistry; expressed the idea of the electromagnetic 
nature of light. According to Albert Einstein, the field idea was Faraday's most original 
idea, the most important discovery since Newton. 
5 James Clerk Maxwell (1831 –1879) was a Scottish physicist. Maxwell laid the foundations of modern classical electrodynamics and introduced the concepts of displacement current and electromagnetic field into physics.  
7 

The course of general physics is traditionally divided into three main 
sections, determined by the nature of the phenomena being studied, the methods of their description, and the models used. These are the sections “Mechanics and Molecular Physics”, “Electricity and Magnetism”, “Optics”, and 
“Atomic and Nuclear Physics.” 
The section “Electricity and Magnetism” is devoted to the study of the 
properties of electric and magnetic fields; here we can distinguish the subsections “Electrostatics”, “Direct Current”, “Magnetic Field”, and “Electromagnetic Induction.” 
The textbook “Electrostatics” consists of four chapters. The first chapter gives the basic concepts and laws of electrostatics: the properties of 
the electric charge, the characteristics, and a graphical representation of the 
electric field using field strength vector lines and equipotential surfaces. Coulomb’s law and the Ostrogradsky–Gauss theorem are reviewed.  
The behavior of conductors and dielectrics in an electric field is discussed in the second and third chapters. The phenomenon of electrical induction and the principles of protection against electrostatic fields are described. 
The concept of the electrical capacitance of an isolated conductor and capacitor is given, and the rules for determining the electrical capacitance of a capacitor bank are given.  
The field of an electric dipole is considered, and the concept of 
the electric moment of a dipole is introduced. The phenomenon and mechanisms of polarization of dielectrics are considered. The concepts of dielectric 
susceptibility and permeability are introduced, and the Ostrogradsky–Gauss 
theorem for the field in dielectrics is formulated.  
The fourth chapter is devoted to the topic of the energy of charged 
bodies in a field, the energy of an electric dipole in an electric field, the energy of a charged capacitor, and the energy of the electric field itself. 
For better assimilation of the material, each chapter contains a detailed 
analysis of typical problems on the topics covered and ends with control 
questions for self-testing. 
At the end of the tutorial, there is a list of textbooks and problem books 
in English recommended for independent extracurricular work.  
8 

.  E L E C T R I C  F I E L D  
An electric field is a type of matter that provides interaction between 
objects that have electric charges. The sources of the electrostatic field are 
electric charges at rest. The opposite statement is also true: electrostatic fields 
are detected by their effect on electric charges at rest. 
1 . 1 .  E l e c t r i c  c h a r g e  
Electric charge is a property of material objects that leads to electromagnetic interactions. Electrostatics considers stationary charges and the stationary electric fields associated with them.  
Let us list the properties that an electric charge has.  
1. There are two types of charges in nature. The existence of two types 
of electricity, “glass” and “resin”, was established in 1734 by the French 
physicist DuFay1. The first appears on glass, rock crystal, precious stones, 
hair, and wool; the second appears on amber, silk, and paper. DuFay established that homogeneous electricity repels and dissimilar electricity attracts. 
In modern terms, two kinds of electric charges (positive and negative) 
exist. Like charges repel each other, and unlike charges attract each other 
(Fig. 1.1). 
 
Figure 1.1. Two kinds of electric charges 
2. Electric charge is a discrete quantity. Any macroscopic electric 
charge q of any physical body is a multiple of the elementary charge e:        
q
Ne
= 
.                                             (1.1) 
                                                          
 
1 Charles Francois DuFay (1698‒1739) was a French physicist. Research in the fields 
of electricity, optics, liquids, heat, and magnetism. 
9