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Файл:Steady electric current. Tutorial
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The Ministry of Science and Higher Education of the Russian Federation
Kazan National Research Technological University
V. Arkhipov
STEADY ELECTRIC CURRENT
Tutorial
Kazan
KNRTU Press
2025
UDC 537.3(075)
Published by the decision of the Editorial Review Board
Arkhipov V.
Steady electric current : tutorial / V. Arkhipov; The Ministry of Education
and Science of the Russian Federation, Kazan National Research Technological University. – Kazan : KNRTU Press, 2025. – 88 p.
ISBN 978-5-7882-3594-3
ISBN 978-5-7882-3594-3
©
Arkhipov V., 2025
©
Kazan National Research Technological
University, 2025
UDC 537.3(075)
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
The main questions of the section “Steady electric current” of the general physics
course are considered. Contains theoretical material on the topics: electric current, electromotive force, Ohm's law, resistance of conductors, Kirchhoff's rules, current power
and the Joule-Lenz law, current in electrolytes, gases and in a vacuum, classical electron
theory of metals and elements of band theory, as well as analysis of solutions to typical
problems, questions for self-testing.
It is intended for students studying Bachelor programs 18.03.01 “Chemical technology” of all specialties of mechanical and technological profiles in order to consoli-
date the theoretical foundations of the course “General Physics”.
It is prepared by the Department of Physics.
2
C O N T E N T S
INTRODUCTION ................................................................................................. 4
1. BASIC CONCEPTS AND DEFINITIONS .................................................... 11
1.1. Electric current ..................................................................................... 11
1.2. Electromotive force, potential difference, voltage .............................. 12
1.3. Steady current and current density vector ............................................ 16
1.4. Problems ............................................................................................... 19
Self-check questions ................................................................................... 22
2. LAWS OF STEADY CURRENT ................................................................... 24
2.1. Ohm's law ............................................................................................. 24
2.2. Branched electric circuits, Kirchhoff's rules ....................................... 29
2.3. Series and parallel connection of resistors and current sources .......... 34
2.4. Thermal effect of current, Joule–Lenz law .......................................... 36
2.5. Temperature coefficient of resistance of metals ................................. 39
2.6. Problems ............................................................................................... 40
Self-check questions ................................................................................... 45
3. CLASSICAL THEORY OF METALS CONDUCTIVITY ........................... 47
3.1. Ohm's law in the classical electron theory of metals ........................... 49
3.2. Joule–Lenz law in the classical electron theory of metals ................... 50
3.3. Problems ............................................................................................... 51
Self-check questions ................................................................................... 55
4. ELECTRIC CURRENT IN OTHER MEDIA ................................................ 57
4.1. Electric current in semiconductors ....................................................... 57
4.2. Electric current in electrolytes ............................................................. 60
4.3. Electric current in gases ....................................................................... 64
4.4. Electric current in a vacuum ................................................................ 68
4.5. Problems ............................................................................................... 71
Self-check questions ................................................................................... 76
5. ELEMENTS OF THE BAND THEORY OF SOLIDS .................................. 77
5.1. Energy bands ........................................................................................ 78
5.2. Metals in band theory ........................................................................... 79
5.3. Dielectrics in band theory .................................................................... 80
5.4. Intrinsic semiconductors in band theory .............................................. 81
5.5. Extrinsic semiconductors in band theory ............................................. 83
CONCLUSION ................................................................................................... 86
BIBLIOGRAPHY ............................................................................................... 87
3
“We will make electricity so cheap that only the
rich will burn candles”
Thomas Alva Edison
I N T R O D U C T I O N
The development of ideas and the discovery of the laws of electric
current would have been impossible without Luigi Galvani1 and Alessandro
Volta2, who created the first sources of steady electric current.
Luigi Galvani, observing the contractions of the muscles of dissected
frogs, explores the physiological effects of electric current and discovers
(1791) “animal” electricity produced by living organisms. His work forms
the basis of electrophysiology. Galvani discovered the emergence of a potential difference upon contact between different types of metal and electrolyte.
Alessandro Volta repeats Galvani's experiments and invents the first source
of electric current – the voltaic pile. The voltaic pile (fig. I) consisted of
several dozen superimposed round plates of silver and zinc or copper and tin,
between which cloth pads soaked in salt water were laid. When dissimilar
metals come into contact, a chemical reaction occurs, accompanied by the
transfer of charges between them and electric current appears.
The voltaic pile is the world's first source of electric current, and is the
ancestor of modern accumulators and batteries. Devices in which the energy
of oxidation-reduction chemical reactions is converted into electrical energy
are called galvanic cells in honor of Luigi Galvani.
The creation of the voltaic pile marked the beginning of intensive research into the properties of electric current.
1
Luigi Galvani (1737–1798) was an Italian physician, anatomist, physiologist and
physicist, one of the founders of electrophysiology and the study of electricity, the
founder of experimental electrophysiology. He was the first to study electrical phenomena during muscle contraction (“animal electricity”). He discovered the occurrence of
a potential difference when different types of metal and electrolyte come into contact.
2
Alessandro Volta (1745–1827) was an Italian physicist, chemist and physiologist, one
of the founders of the doctrine of electricity, inventor of a source of steady electric current. With this invention Volta proved that electricity could be generated chemically and
debunked the prevalent theory that electricity was generated solely by living beings.
4
Fig. I. Schematic diagram of a copper–zinc voltaic pile. Each copper–zinc
pair had a spacer in the middle, made of cardboard or felt soaked
in salt water (the electrolyte)
Research on the chemical effect of steady current begins: Nicholson1
and Carlyle discover the phenomenon of electrolysis (1800) and separate water into hydrogen and oxygen using electric current; Humphry Davy2 decomposes caustic alkalis with electric current and discovers new elements – potassium, sodium, calcium.
Summarizing the experimental material, Faraday3 proves the identity
of the then known types of electricity: “animal”, “magnetic”, thermoelectric-
ity, electricity arising from friction, galvanic electricity and establishes the
laws of electrolysis (1833).
Research on direct current circuits is carried out by Georg Ohm4.
He experimentally established (1826) the basic law of an electrical circuit
connecting current, voltage and resistance. Ohm introduces the concepts of
1
William Nicholson (1753–1815) was an English physicist, chemist and engineer. Together with Anthony Carlyle (1768–1840), he constructed the first electric battery in
England and discovered the phenomenon of decomposition of water by electric current.
2
Humphry Davy (1778–1829) was an English chemist and physicist. He proved that
electric current causes the decomposition of acids and salts. In 1821 he established the
dependence of the resistance of a conductor on its length and cross-section, and observed a change in resistance with a change in temperature.
3
Michael Faraday (1791–1867) was an outstanding English experimental physicist and
chemist. Work in the field of electricity, magnetism, magnetooptics, electrochemistry.
4
Georg Ohm (1787–1854) was a German experimental physicist, theoretically derived
and experimentally confirmed a law expressing the relationship between current
strength in metal conductors, voltage and resistance.
5
“electromotive force”, “voltage drop”, “electrical conductivity” and deter-
mines the dependence of the current in a metal conductor on its length, crosssection and voltage at the ends of the conductor. The unit of electrical resistance (Ohm) is named after him.
Electrical measuring instruments are being improved. The first galvanometers1 appeared in the early 20s of the 19th century. Poggendorff's galvanometer2 consisted of a wire coil containing a magnetic needle inside. For accurate measurements, the mirror reading method was used. In 1837, Poulier3
invented a device for measuring current strength, called the tangent galvanometer or tangent compass (fig. II). A small magnetic arrow equipped with a long
copper pointer, mounted on a needle, was placed in the center of a vertical
wire ring (40–50 cm in diameter) with the current being measured.
Fig. II. Tangent galvanometer or tangent compass
1
Galvanometer (from the surname Luigi Galvani and the Greek μετρέω – measure) is
a highly sensitive device for measuring small direct electric currents.
2
Johann Poggendorff (1796–1877) was a German physicist, works in the field of electricity and magnetism, history of physics. In 1821 he invented the galvanometer, in 1826
he proposed the method of mirror reading in electrical measuring instruments, and in
1842 he designed a rheostat, developed a compensation method for measuring small
electromotive forces.
3
Claude Poulier (1790–1868) was a French physicist, works in the field of heat, optics,
electromagnetism.
6
7
Methods for measuring the resistance of conductors are being developed. Wheatstone1 in 1843 invents a bridge circuit for precise measurement
of resistances (Wheatstone bridge). Kirchhoff2 in 1845–47 established regularities in the flow of electric current in branched electrical circuits (Kirchhoff's rules).
Jacobi3 in1848 proposed a unit of resistance, made a standard of resistance and sent it to a number of scientists. The Jacobi resistance standard
was a coil of uniform copper wire 25 feet (7.61975 m) long and weighing
345 grains (22.4932 g). The coils were placed in wooden boxes and filled
with an insulating compound.
The thermal effect of current is studied by the English scientist Joule4
and the St. Petersburg academician Lenz5. Independently of each other, they
establish a law that determines the amount of heat released by electric current
in a circuit—the Joule–Lenz law (1843–44).
Research on the phenomenon of thermoelectricity is being carried out.
Seebeck6 (1821) noticed that the connection of dissimilar metals whose contacts have different temperatures causes a deflection of the magnetic needle.
He built a thermocouple and used it to measure temperature. In 1834, Peltier7
discovered that when current is passed through a circuit of two different
1
Charles Wheatstone (1802–1875) was an English physicist, work in the field of elec-
tromagnetism, acoustics, optics.
2
Gustav Kirchhoff (1824–1887) was a German physicist. The works are devoted to
electricity, mechanics, optics, thermal radiation (Kirchhoff's law), mathematical physics, elasticity theory, hydrodynamics. It was he who introduced the concept of an absolutely black body into physics.
3
Boris Semyonovich Yakobi–Moritz Hermann von Jacobi (1801–1874) , was a German
and Russian Imperial engineer and physicist. Jacobi worked mainly in the Russian Empire. He furthered progress in galvanoplastics, electric motors, and wire telegraphy.
4
James Prescott Joule (1818–1889) was an English physicist, mathematician. Joule
studied the nature of heat, and discovered its relationship to mechanical work. This led
to the law of conservation of energy, which in turn led to the development of the first
law of thermodynamics. The SI unit of energy (Joule) is named after him.
5
Emil Khristianovich Lenz (1804–1865) was a Russian physicist who is most noted for
formulating Lenz's law in electrodynamics in 1834. Besides the law named in his honor,
Lenz also independently discovered Joule's law in 1842; to honor his efforts on the
problem, it is also given the name the “Joule–Lenz law”.
6
Thomas Johann Seebeck (1770 –1831) was a German physicist. The works are devoted
to electricity, magnetism, and optics.
7
Jean Charles Peltier (1785 –1845) was a French physicist, author of numerous papers
in different departments of physics.
8
metals, heat is released or absorbed at the junctions, depending on the direction of the current (Peltier effect).
Electric current finds practical application in telegraph and telephone
communications. The first electromagnetic telegraph was created by the Russian scientist Schilling1 in 1832. Subsequently, the electromagnetic telegraph
was built in Germany by Gauss2 and Weber3 (1833), in Great Britain by Cook
and Wheatstone (1837), and in the USA the electromagnetic telegraph was
patented by Morse4 in 1840.
Electric lighting devices were invented, the arc lamp of Yablochkov5
(1876), the incandescent lamp of Lodygin6 (1874) and Edison7 (1879). Yablochkov wrote “... electric lighting spread throughout the world, reaching the
palace of the Shah of Pesidia and the palace of the King of Cambodia”.
Electric current generators are created – dynamos. The first generator
that produced electricity on an industrial scale was invented by Gramme8.
Electric motors operating on direct current appear. The dynamo machine
became the prototype from which further inventions emerged, such as the direct
current (DC) motor, the alternating current (AC) generator, the synchronous
motor, and the rotary converter AC to DC, or DC to AC power.
1
Pavel Lvovitch Schilling (1786–1837), also known as Paul Schilling, was a Russian
inventor. He developed the first electromagnetic telegraph that was of practical use.
2
Carl Friedrich Gauss (1777–1855) was a German mathematician, astronomer, geode-
sist, and physicist who contributed to many fields in mathematics and science.
3
Wilhelm Eduard Weber (1804–1891) was a German physicist. The works relate to the
field of magnetic phenomena and electricity. The unit of magnetic flux (Weber) is
named after him.
4
Samuel Finley Morse (1791–1872) was an American inventor and painter. The most
famous inventions are the electromagnetic recording telegraph (Morse apparatus) and
the Morse code (alphabet).
5
Pavel Nikolaevich Yablochkov (1847–1894) was a Russian electrical engineer, military engineer, inventor and entrepreneur. He is known for developing an arc lamp, which
went down in history as the “Yablochkov candle”.
6
Alexander Nikolaevich Lodygin (1847–1923) was a Russian electrical engineer, one
of the inventors of the incandescent lamp, entrepreneur. He was the first to propose
using tungsten filaments in lamps and twisting the filament into a spiral.
7
Thomas Alva Edison (1847–1931) was an American engineer, inventor and businessman. Creator of the phonograph; improved the telegraph, telephone, motion picture
equipment, developed one of the first commercially successful versions of the incandescent electric lamp.
8
Zenob Theophile Gramme (1826–1901) was a Belgian electrical engineer. He invented
the direct current dynamo, which could generate high voltage. In 1878, Gramme's machines were first used to light Paris.
9
J. J. Thomson1 in 1897 while studying the effect of magnetic and electric fields on cathode rays (fig. III) discovers the electron. A new stage in the
development of the physics of electromagnetism begins.
Fig. III. Schematic diagram of Thomson's experiment with cathode rays:
C is the cathode, A is the anode, B is the focusing slit, D and E
are the plates between which an electric field is created
Thomson's experiments showed that cathode rays are deflected by
both an electric and a magnetic field, and represent a stream of charged particles with a speed much lower than the speed of light.
In the works of Thomson, Drude2 and Lorentz3 (1898–1904), the classical electron theory of metals was created, based on Ricke4 concept of the
“electronic gas”. Conduction electrons in a metal behave like molecules of
an ideal gas. Unlike gas molecules, whose range is determined by the collisions of molecules with each other, electrons collide primarily not with each
other, but with the ions that form the crystal lattice of the metal.
Edison's discovery of the phenomenon of thermonic and thermoelectronic emission (1880) formed the basis for the operation of electronic tubes:
a two-electrode electronic tube (diode), invented by Fleming5 (1904), a three-
1
Joseph John Thomson (1856–1940) was an English physicist. His works are devoted
to the study of the passage of electric current through rarefied gases, cathode and
X-rays. Nobel Prize laureate in physics (1906). He proposed (1904) one of the first
models of the atom—the plum pudding or cake model.
2
Paul Karl Ludwig Drude (1863–1906) was a German physicist specializing in optics.
He wrote a fundamental textbook integrating optics with Maxwell's theories of electromagnetism.
3
Hendrik Antoon Lorenz (1853–1928) was a Dutch theoretical physicist, works in the
field of electrodynamics and optics. Nobel Prize laureate in physics (1902).
4
Eduard Riecke (1845–1915) was a German experimental physicist. He conducted
a large number of experiments on the conductivity of electricity in metals and gases,
ferromagnetism, crystallography, thermo- and hydrodynamics, thermodynamics, physical chemistry, piezo- and pyroelectricity, and phenomena in cathode ray tubes.
5
John Fleming (1849–1945) was an English physicist, works in the field of radiotelegraphy, radiotelephony, radio and electrical engineering.
10
electrode electronic tube, triode (fig. IV) invented by Forest1 (1908). With
the advent of electronic tubes, radio electronics arose. One of the symbols of
scientific and technical progress in the 20th century is precisely the electronic
tube.
Fig. IV. Three-electrode vacuum tube (audion) invented by de Forest
While studying the electrical properties of substances at extremely low
temperatures, Kamerlingh Onnes2 obtained liquid helium (1908) and discovered (1911) the phenomenon of superconductivity.
Revolutionary changes in physics are associated with the ideas of
quantization of the properties of micro-objects. In 1926, Schrödinger3 creates
wave or quantum mechanics. In 1928, Sommerfeld4 develops the quantum
theory of metals, a quantum-mechanical model (the Drude–Sommerfeld
model) of the behavior of charge carriers in a metalic solid, combining the
classical Drude model with quantum-mechanical statistics. The band theory
of solids arises, which explains the electrical conductivity of metals, semiconductors, and dielectrics based on unified concepts of allowed and forbidden energy zones.
1
Lee de Forest (1873–1961) was an American inventor, electrical engineer, and an early
pioneer in fundamental electronics. He invented the first triode, consisting of an evacuated glass tube containing three electrodes: a heated filament (the cathode, made of tantalum), a grid, and a plate (the anode). He had over 300 patents worldwide.
2
Heike Kamerlingh Onnes (1853–1926) was a Dutch physicist. Nobel Prize in Physics
for his investigations of the properties of bodies at low temperatures and the production
of liquid helium, 1913. Kamerlingh Onnes earned the honorary nickname “Mr. Absolute
Zero” from his colleagues.
3
Erwin Schrödinger (1887–1961) was an Austrian theoretical physicist, one of
the founders of quantum mechanics. Nobel Prize laureate in physics (1933).
4
Arnold Sommerfeld (1868–1951) was a German theoretical physicist. His works are
devoted to the quantum theory of the atom, spectroscopy, quantum theory of metals,
mathematical physics.
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