Electrodynamics, Maxwell's equations

Chapter 1
GRAVITATION AND ITS PLACE IN PHYSICS
1.1. Development of knowledge about nature, universe, gravity
1.2. Newton's law of gravitation
1.3. Mechanical LeSage hypothesis of the nature of gravity;
1.4. Analogy between gravity and electrostatics
1.5. Electricity and magnetism. Maxwell's equations. Electromagnetic waves.
1.6. Four-dimensional spacetime and special theory of relativity

1.5. Electricity and magnetism. Maxwell's equations. Electromagnetic waves.

The most important force that determines all internal structure and behavior of natural objects, from subnuclear, atomic, and molecular scales, to the macroscopic dimensions of surrounding nature (including ourselves) and the scales of the Earth and other planets, is electromagnetic interaction. Carriers of electric forces are the basic building blocks of atoms - electrons carrying a negative elementary electric charge and protons carrying a positive charge (positive and negative signs evolved by convention). The electric forces between protons and electrons, in co-production with quantum laws, determine the structure of atoms, and thus the chemical and physical properties of substances (... "Interaction of atoms" ...).
  Each electric charge (charged body) excites an electric field around it according to Coulomb's law (1.20b) with an intensity proportional to the magnitude of the charge and inversely proportional to the square of the distance; if the charge does not move (in the given reference system), it is an electrostatic field. The electric field exerts force effects on every other charged body that enters this space. If the charge moves (it is an electric current), in addition to the electric field, it also excites a magnetic field according to Biot-Savart-Laplace's law (1.33a). The magnetic field shows force effects on each electrically charged body that moves perpendicular to the direction of the magnetic field vector (Lorentz force). The combination of both fields represents an electromagnetic field. When electric charges move at a variable speed (with acceleration or deceleration), they create a time-varying electromagnetic field around them, which leads to the formation of electromagnetic waves that detach from their source and carry some of its energy into space. According to Faraday's law, electromagnetic induction of an electric field arises when motion or time changes in a magnetic field; and temporal changes in the electric field in turn cause a magnetic field. This field is governed by Maxwell's equations of the electromagnetic field, which were created by combining and generalizing all the laws of electricity and magnetism. The combined science of electricity and magnetism, including the dynamics of charge motions and the time variability of fields, is called electrodynamics. This will be the main content of the existing §1.5.
  In the previous §1.4 we saw that the analogy between Newton's gravistatics and Coulomb's electrostatics is very tight. However, the electrostatic field is a special case of the general electromagnetic field that prevails in the vicinity of moving electric charges. It is therefore useful to note the properties of the electromagnetic field and to try to find possible analogies with the general "gravidynamic" field around moving bodies. Electrodynamics is the most perfect and successful theory of classical physics, which retains its full validity even in modern relativistic physics. It can be said that electrodynamics is one of the cornerstones of all physics and has played a key role in shaping the special and general theories of relativity, as well as quantum physics.
Note: The historical development of knowledge about electricity and magnetism is briefly outlined in §1.1 in the passage "Electrodynamics, atomic physics, theory of relativity, quantum physics". The relativistic view of the relationship between electric and magnetic fields is briefly discussed below in the section "Relativistic Electromagnetism".
Linearity of electromagnetism :
Electric and magnetic action in vacuum is linear in sources (electric charges of different sizes) and in the values of fields excited directly or by induction. The principle of superposition applies here exactly. The values of electric and magnetic field intensities from different sources are simply added up (vectorially). This is no longer entirely true in the material environment, where the effects of interactions of electric and magnetic fields with the configurations of atoms in materials are applied. Below we will see that this can occur when an electric field is excited by charges in inhomogeneous dielectrics, at their interfaces, and especially when a magnetic field is excited in ferromagnetic substances, where the saturation state also manifests itself.
  In quantum field theory, higher-order effects occur when photons interact through fermion loops. We leave aside here the hypothesis of nonlinearity of electrodynamics at superstrong field intensities ("Nonlinear electrodynamics")...

Physical units in electricity and magnetism
During the long-term development of physics and natural science, not only were new knowledge constantly acquired, but also various physical quantities and units for their quantification were introduced. Particularly dynamic development and abundance of various units took place in the field of electricity and magnetism during the late 18th to the first half of the 20th century.
  Systematic work on the creation of a unified and rational system of physical units began in 1862 on the initiative of the British Association for the Advancement of Science. Thus, the "absolute" CGS system was created based on three basic units: Centimeter, Gram, Second. The International Conference on Weights and Measures was established and etalons for the meter and kilogram were implemented. However, there were many different electrical and magnetic units, often based on secondary empirical knowledge (e.g. the Ampere unit based on the decomposition of silver by electric current during electrolysis, or the Ohm using the electrical resistance of a mercury column). From the point of view of the nature and connection of phenomena, the CGSE system proposed by C.F. Gauss is often used in fundamental (theoretical) physics, which assumes the coefficients e₀ and m₀ to be equal to 1 - it includes them in the quantities of electric E and magnetic B field intensities. This leads to concise and concise notations of equations between electromagnetic quantities. We will mostly use them in theoretical analyses. However, we will also present important resulting formulas and practical applications in SI units.
  The international system of units SI developed mainly in the first half of the 20th century. The three basic mechanical units meter, kilogram and second were supplemented by the electrical unit ampere in 1950 - thus the MKSA system was created. In 1960, the International Conference on Weights and Measures named the system based on the meter, kilogram, second, ampere, kelvin and candela "Systeme international d'unités" (International System of Units) with the abbreviation SI.
  For technical applications, the SI system is now the most practical and almost exclusively used, because its units are mostly close in size to the dimensions and intensities in common nature; and also because all current measuring instruments are calibrated in these units.
Note:  Incorrect or illogical definition of the basic units of electromagnetism !
The historical development of the knowledge of basic physical laws sometimes took quite convoluted paths. Along with this, definitions of physical quantities and their units were also formed, which were indebted to the ideas of the time.
  The introduction of electric current as a basic quantity and its SI unit Ampere (using the "magnetic force action of two infinite parallel conductors...") was unfortunate. Physically, the primary thing is the electric charge, from which the electric current should be derived as the amount of charge passed per unit of time (Coulomb/second). Or in magnetism, the terminological illogicality of the names "magnetic field intensity" and "magnetic induction" - it should be the other way around! - (for an electric field, it is fine). It is briefly discussed below in the passage "Intensity<->Induction in electromagnetism". This unfortunate "crossing" of the names "intensity-induction" arose during the historical development of the science of electricity and magnetism, when magnetism was mistakenly explained by fluid theory, analogous to electrostatics. And unfortunately, it has remained so in the current SI system ...

Integral and differential formulations of physical equations
Electrodynamics is a branch of physics where derivatives and integrals of vector fields are used most in mathematical formulas, often unified into the formalism of differential operators. Fellow physicists are certainly well acquainted with these techniques, but for those interested in other fields, I would like to try to write a few notes here about their essence and benefits. Physical fields are expressed by the values f of forces-intensity and energies-potentials at different locations in space. They are therefore functions of location - of spatial coordinates x,y,z and time t - f(x,y,z,t). We will first show this on functions of one variable f(x) and their graphical representation.
  To analyze natural processes in various situations, it is often necessary to find out how quickly (how sharply, with what gradient) one parameter changes in relation to another parameter. This is quantified by the mathematical operation of the derivative of one parameter with respect to another parameter - it is written ^{d f(x)}/_dx| _x=x0 at the point x₀, it is also denoted f '(x). It is the steepness of the change in the value of this function with respect to an infinitesimal change in its independent variable. In mechanics, the derivative of position with respect to time expresses the speed of movement of a body. For a function of one variable f(x), the derivative determines the slope of the tangent to the curve of its graph at a given point x₀. For functions of several variables f(x,y,z,t), partial derivatives ¶f/¶x, ¶f/¶z, ¶f/¶z, ¶f/¶t are introduced - partial gradients in the direction of individual coordinate directions x,y,z and time t. Here, only the variable with respect to which the derivative is made is taken as a variable, while the other variables are considered constant.

Derivative of the function f(x)	Integration of the function f'(x)

We often also need to add-sum the local instantaneous values of a certain quantity into the resulting accumulated value, which can determine the functional behavior of other quantities. If these are constant values, it is a simple arithmetic operation of "+" addition. However, with variable values of the function f(x), this addition must be performed gradually, locally as integration. It is written nf(x) dx. The integral sign n is a vertically extended band "S", an abbreviation for summation. The integrated range x₁÷x₂ is divided into infinitesimal sections dx and the elementary products f(x1+n.dx).dx are gradually added until x₂ is reached. The integral of the function f(x) gives the area under its curve between the values x₁ and x₂.
  In a two-dimensional generalization, for functions of two variables f(x,y), integration is done using infinitesimal parallelograms and the area integrals _Snnf(x,y) dS of either scalar quantities (such as mass or charge using their areal density) or the flux of a vector field over some given area S are computed. Next, curve integrals are computed along some parameterized curve in the 2-D plane. Area integrals of vector functions can be converted to volume integrals using Gauss's divergence formula, or to line integrals using Stokes's rotation formula - see the figure below.
 Differentiation and integration of functions
Above, we outlined the derivative of the function f(x) at a specific point x₀ and its integral in a certain range x₁-x₂. The result is a certain local number. However, for a more complete analysis of the behavior of functions, it may be important to perform the differentiation and integration operations at all locations of the function f(x) - for all values of the variable x (in a given domain of definition). The result of this process is then a new function f '(x) or F(x), which shows the differential or cumulative trend of the original function f(x).
  By differentiating the function f(x), the function f '(x) is created, which quantifies the variability of the original function f(x) at each location when the independent variable x changes. In places where the function f(x) is increasing, the derived function f '(x) is positive and its value is proportional to the steepness of the growth, in decreasing regions of f(x) the derivative goes to negative values. Where f(x) is constant, or has local maxima or minima, the derivative f '(x) is zero.
  The integral nf(x) dx = F(x) is called an indefinite integral, because it has no specified limits for the independent variable x, it is integrated continuously over the entire domain of the function f(x). If the function f(x) is nonnegative, its integral F(x) is a monotonically increasing function. If the function f(x) also goes to negative values, negative values may also prevail even in the integrated function F(x).

Differentiating the function f(x) gives the function f '(x)	Integrating the function f(x) gives the function F(x)

Differentiating and integrating functions are mutually opposite processes - from the derived function f ´(x) we can obtain the original function f(x) by integrating, using the initial condition. Conversely, from the function F(x) we can obtain the original function f(x) by differentiating. The function F(x) is sometimes called the "primitive function" of the function f(x).

Differential operators
Derivatives of vector field functions (here electric and magnetic field intensities) are combined-unified into the formalism of the so-called differential operators for better clarity :
Nabla N: The basic differential operator here is "nabla N": NF = ¶F/¶x + ¶F/¶y + ¶F/¶z . Other derivative combinations are then derived from the operator N :
Gradient: grad F = NF = [¶F/¶x, ¶F/¶y, ¶F/¶z] quantifies the steepness of changes in the scalar field F at different locations. In electrodynamics, this is the gradient of the potential f.
Divergence of the vector function F: div F = N . F = ¶F_x/¶x + ¶F_y/¶y + ¶F_z/¶z. It quantifies the local flow - divergence, convergence - of the vector field F. In electrostatics it expresses the way in which the distribution of electric charges creates the electric field E - (1.32b).
Laplace operator: D f = div grad f = N² f = ¶²f/¶x² + ¶²f/¶y² + ¶²f/¶z². It quantifies the dynamics of the change of the field F in space. In the 4-dimensional formulation x,y,z, c.t of the special theory of relativity, the d'Alembert differential operator š f = ¶²f/¶x² + ¶²f/¶y² + ¶²f/¶z² - (1/c2)¶²f/¶t² is used.
Rotation or curl is the vector product of the operator nabla N and the investigated vector function F: rot F = N × F = [(¶F_z/¶y ^- ¶F_y/¶z)+(¶F_x/¶z ^- ¶F_z/¶x) + (¶F_y/¶x ^- ¶F_z/¶z)]. It quantifies the local rotation - circulation - turning - of the vector field, changes in the direction of the vector F in the vector field. It is expressed by the differences of the partial derivatives F_x,y,z between the coordinates x,y,z. It is very well suited for modeling the magnetic field B, which has a circular-spiral shape around the exciting moving charges or fluxes - (1.33)-(1.37).
  The connections between the differential relations and the integral dependencies of physical quantities, here the field intensities and potentials, are important here. For the differential operators "div" and "rot" two important integral equations hold :
--» The Gauss-Ostrogradsky divergence formula shows that the area integral of the vector field F(x,y,z) over a closed surface S is equal to the volume integral of the divergence of the field div F over the volume V inside this closed surface. This means that the flux of the vector field F over a closed surface S is equal to the volume integral of the divergence of the field div F, i.e., the local increments and decrements of the field F, in the inner region enclosed by the surface S.
--» The Stokes integral rotation formula shows that the flux of the vector rot F through the surface S in space is equal to the curve integral of the circulation of the vector F along the curve C that bounds this surface. We can imagine this as the local rotations of the vector field F on the surface S are added to the resulting circulation of the vector field F along the total curve C bounding the surface S. For the magnetic intensity vector B see formula (1.34.b), (1.37.b).

Gauss-Ostrogradsky divergence formula	Stokes integral rotation formula

Electric charge
The name "electric charge" is used in electrodynamics in two meanings :
1. A body or particle that exhibits a force of electrical action. We also say that it is a carrier of electric charge. They are primarily electrons and protons, and then ions and bodies that have a mutual excess of electrons or protons.
There are also other charged particles in the microworld - muons, pions, hyperons (§1.5 "Elementary particles and accelerators", part "Elementary particles and their properties"), which, however, are very unstable, do not occur in our nature and have no significance for the science of electricity importance.
2. A physical quantity that quantizes the size - the measure of electric charge. The basic unit of charge is 1 Coulomb. In atomic and nuclear physics, the electron charge 1 e = 1.602x10^-19 Coulomb is also often used as a unit.
  In field theory, the distribution of electric charges is expressed by the charge density r(x, y, z, t), which is generally a function of place and time, so that the total charge contained in the spatial region V is Q = _V òòò r dV.
  In an electromagnetic field acts on a test particle with a charge q moving at a velocity v the total force (Lorentz force)

where E is the intensity of the electric field and B is the intensity of the magnetic field (for historical reasons called magnetic induction), "x" means the vector product. Below, we will first discuss the origin and properties of electric and magnetic fields separately, and then their mutual connections and dynamic behavior in the electromagnetic field.

Movement of electric charges - electric current
In the science of electricity, the movement of electric charges is generally called an electric current. Of particular importance is the orderly movement of charges, especially in conductors. In a narrower sense, the ordered movement of electric charge carriers is therefore called an electric current. It is quantified by the electric current I, which is the electric charge q passing through the cross-section of the conductor per unit of time: I = dq/dt. The unit in the SI system is 1 Ampere, which is the charge of one Coulomb passed in 1 second (the awkward technical definition of 1A using the "force action of two infinite parallel conductors" is not important to us).
  According to the type and movement of the charge carriers, the electric current is divided into two basic groups :
-> Conductive - drive current is an ordered flow of free charge carriers in a material environment under the action of an electric field. Above all, it is the movement of free electrons in metal conductors, the movement of ions in electrolytes or in gases during electric discharges. Particles carrying an electric charge collide with atoms of a substance as they move through the medium, transferring part of their kinetic energy to them and causing them to oscillate. This results in losses of electric current energy and heating of the medium. A conductive medium offers a certain resistance to the electric current (minimization or almost nullification of resistance is discussed in §..., passage "superconductivity").
-> Convective - flowing electric current caused by the mechanical movement of charge carriers in the environment, without the instantaneous effect of an electric field (the charge carriers are either carried by the flowing material medium, or move by inertia in a vacuum). An important example of convection current is the movement of charged particles in accelerators. In a convection current, there are no collisions of charged particles with particles of the environment, so there are no thermal effects, but only electric and magnetic ones.
  In terms of the time course and direction of charge flow, we encounter two types of electric current :
-> Direct current, in which electric charges do not change the direction of their flow over time. The magnitude of the current can be either constant over time (during the monitoring period of the function), or variable - increasing, decreasing, pulsating (while maintaining the same direction). The source of direct current is, for example, electrochemical galvanic cells and accumulators, thermocouples, photovoltaic cells. Common electronic sources are rectifiers that obtain direct current from alternating current.
-> Alternating current, which periodically changes the direction of its flow over time. The periodic waveform can be different, for example, rectangular (simple alternation of "+" and "-"), sawtooth, but the most common is sinusoidal - harmonic: I(t) = I_max . sin(w.t + j), where I_max is the amplitude, w is the angular frequency related to the frequency f by the relation w = 2p.f a j (0÷360°) is the phase shift of the beginning of the time coordinate t (or the phase shift between voltage and current). The frequency f indicates the number of oscillations per unit of time; the unit 1 Hz means one oscillation per 1 second (the name is after one of the pioneers of electromagnetism H.Hertz). Opinions on what is low-frequency or high-frequency differ, depending on the field in which alternating current is used. In everyday life and in electroacoustics, 20 kHz is usually taken as the limit. In radio engineering, this limit is usually moved up, to the MHz range...
We basically have two types of alternating current sources :
--» Alternators are rotary power electro-mechanical sources of alternating current for energy needs. The source of mechanical energy is a rotating turbine - steam in thermal and nuclear power plants, or water in hydroelectric power plants (or a propeller in a wind source). The turbine drives the magnetized rotor of the alternator, which creates a rotating magnetic field. An alternating voltage with a frequency given by revolutions/s is then induced in the stator coils. Alternators in power plants have three coils wound in their stator, angularly offset by 120°, thus creating a 3-phase current. In the globally interconnected electrical network, all alternators operate synchronously with a frequency of 50 Hz, in the USA with a frequency of 60 Hz.
Small and much simpler 1-phase alternators with a voltage of about 14V are used in cars with an internal combustion engine, that drives them and they charge the battery and power the ignition, headlights and other electrical equipment..
--» Oscillators are electronic circuits in which periodic oscillations of electrical voltage and current occur with a frequency dependent on the parameters of the components (capacitors, coils, resistors, transistors) and can often be tuned. Mostly harmonic sinusoidal oscillations are created (exceptionally rectangular or sawtooth in multivibrators) for use in radio engineering or instrumentation (for more details, see "Transmission and reception of electromagnetic radio waves" below).

Electrical components and circuits in electronics
In the practical use of electricity, electric current and voltage, electrical components (elements) with various required properties are used. From the point of view of supplying or consuming electrical energy, we can divide these components into two categories :
-> Active - sources, which supply electrical energy to the circuit. In heavy-current power engineering, these are electro-mechanical generators (alternators, dynamos), which use rotating turbines or propellers to convert the mechanical energy of steam, water or wind into electrical energy. Then there are photovoltaic cells and galvanic cells. The picture on the left shows a schematic representation of the production of electrical energy in a power plant (nuclear here), the transformation of voltage from 6 kV to 220-400 kV, in the middle is the long-distance line to the consumption area. For long-distance energy transmission, it is advantageous to transform to high voltage, where a relatively small current is sufficient, which minimizes ohmic losses in the line and also thinner wires are sufficient (mostly Al-Fe ropes with a cross-section of about 300mm² are used). However, if the voltage is too high, about >500kV, losses increase again due to air ionization and small corona discharges. The picture on the right shows a schematic representation of the transformation to 22 kV in a transformer station and finally to 220 V for powering common electrical appliances at consumers :
        
                                                                      Production and distribution of electrical energy
  In addition to these basic primary sources, there are secondary electronic sources, which use this primarily created electricity to power the resulting electrical circuits in household appliances, industrial machines and laboratory instruments - transformers, rectifiers, converters :

Power supply of electrical circuits

Alternator in a power plant      Power transformer   Instrument transformer  Small transformer    Galvanic battery    Fuses, circuit breakers and switches

-> Passive components, which take electrical energy and convert it into other types of energy (thermal, light, mechanical, sound, electromagnetic...). We will mention a few of the most common electronic components :
=» Resistors, whose task is to provide a certain increased resistance to the electric current, which reduces the electric current and creates a certain voltage drop across the resistor. When a stronger current (e.g. several Amps) passes through the resistor, considerable heat is generated, so resistors can also be used as heating elements. The unit of electrical resistance is 1 Ohm [?]: A resistor has a resistance of 1? when a current of 1 A flows through it at a voltage of 1 V. Resistors are made of conductive materials with increased resistivity, such as alloys of iron, nickel and chromium, or copper and nickel, for higher resistances graphite. Either in the form of metal resistance wires, or thin layers of graphite or metallic or metal oxide layers, deposited on insulating (usually ceramic) carriers with milled grooves to increase the length of the layer and thus increase the resistance. A resistor with a controllable adjustable resistance using a third electrode, mechanically moved along the resistive layer, is called a rheostat or potentiometer (the name comes from the fact that it is a resistive voltage divider previously used in devices for measuring electrical potential, voltage).
=» Capacitors sometimes also called condensers. The basic design of a capacitor consists of two conductive plates (electrodes), separated from each other by an insulating layer of dielectric. When electric charges of opposite polarity are applied to each of the plates and attract each other, the insulating dielectric between them does not allow the charge carriers to come into contact. The plates remain charged even after the electrical source is disconnected. The basic property of a capacitor is to accumulate an electric charge Q. This ability is quantified, in relation to the electric potential - voltage U, by a quantity called capacitance C = DQ/DU. In general, every conductive body has an electric capacitance. The unit of capacitance in the SI system is 1 Farad: A body has a capacitance of 1 Farad if the supply of 1 Coulomb of charge increases its voltage by 1 Volt. 1 Farad is a very large unit, therefore its decimal fractions are used: microfarad 1µF = 10^-6 F, nanofarad 10^-9 F, picofarad 10^-12 F. Common isolated bodies have a very small capacity of about units to tens of picofarads (the capacity of the human body is about 30pF). In capacitors, their increased capacity is caused by the large area of the plates placed in close proximity to each other, where opposite electric charges are strongly attracted to each other. The capacity of a capacitor is given by the size-area of the plates S, their mutual distance d and the permittivity e of the insulating dielectric layer: C = e. ^S/_d . The classic technical design is a scroll capacitor whose electrodes are long thin aluminum strips, between which there is paper or plastic foil, wound into a small cylinder. They are produced in capacities of about nanofarads to tens of microfarads. For higher capacities of tens, hundreds and thousands of microfarads, electrolytic capacitors are used, where an aqueous electrolyte solution (usually boric acid) is contained in a hermetically sealed cylinder, in which an aluminum electrode is immersed. High capacities are achieved here by a very thin dielectric layer and high permittivity of the electrolyte. Conversely, for very small capacities of units and tens of pF, small metal plates in air are simply used. These are also capacitors with variable - tuned - capacitance, where the sheet metal electrodes are inserted into each other by rotational movement. They are used in frequency tuning circuits (see below "Targeted transmission and reception of radio waves"). ...... varicaps ......
=» Induction coils wound from a conductive wire, most often copper. The electric current passing through the winding creates a magnetic field inside. Every conductor, when passing a current, creates a magnetic field according to the Biot-Savart law (1.33b) of excitation of a magnetic field by an electric current. When the current passing through it changes over time, this changing magnetic field is accompanied by electromagnetic induction of voltage according to Faraday's law, and this induced voltage acts against the supply voltage. The inductance L of a conductor is its ability to induce a voltage in itself due to changes in the current flowing through it. The unit (intrinsic *) of the inductance of a conductor in the SI system is 1 Henry (according to J.Henry, who, along with Faraday and Maxwell, was also a pioneer of electromagnetism). A conductor or coil has an inductance of 1 Henry when, when the current changes by 1 Ampere per second, a voltage of 1 Volt is induced in it. The intrinsic inductance of a wire or coil can be simply considered as a kind of "electrical inertia": he defends himself - resists - changes in the current flowing through it, by inducing an opposite voltage..
*) If there is another conductor near this conductor, a certain voltage will also be induced in it due to the variable current of the neighboring conductor. Here we are talking about mutual inductance.
  Even a simple straight conductor has a certain small self-inductance, which depends on the length of the wire and its thickness (longer and thinner wires have a greater inductance L analogously to the resistance R; however, these dependencies are not linear here, since they depend on the spatial course of the magnetic field around the conductor). For a straight wire of circular cross-section, its self-inductance L [nanoHenry] is given by the semi-empirical formula L_[nH] = µ . l . [ln(^2.l/_r) - 1], where l is the length of the wire and r its radius (thickness/2) in meters, m is the relative permeability. For example, a wire 1 meter long and 1 millimeter thick has an inductance of about 1.5 mH.
  In the case of alternating current, a variable alternating magnetic field is created, which in turn induces an electric voltage - self-induction. This is combined with the passing one, acts against it, causing a phase shift between the voltage and the current. The coil presents a certain resistance - impedance - to the alternating current, which depends on the frequency (see below). Coils are wound either "in air" without a core, or around a ferromagnetic core. The shape of the coil axis is either straight - the so-called solenoid, or circular toroidal. A simple solenoid-shaped coil has an inductance L = µ.N² .^S/_l , where S is the cross-sectional area of the coil, µ is the permeability of the medium, N is the number of turns of the coil, l is the length of the coil. If it is wound on a ferromagnetic core, its inductance will increase in proportion to the relative permeability of the core material. Toroidally wound coils are characterized by high inductance and low dispersion of the magnetic field into the surroundings.
=» Transformers are systems of magnetically coupled coils that can convert (transform) alternating current of a certain voltage to a higher or lower voltage using electromagnetic induction, while allowing galvanic separation of both electrical circuits. It consists of two or more coils (windings) electrically isolated from each other, but sharing a common magnetic field :
- The primary winding is called the one to which the initial (supply) alternating electric current or voltage signal U₁ is supplied. This excites an alternating magnetic field.
- In the secondary winding, this variable magnetic field electromagnetically induces an alternating voltage U₂, which is taken from there to another circuit or consumer.
  The magnetic coupling of both windings is realized by both coils being placed close to each other or one inside the other, most often they are wound on a common ferromagnetic core. In the magnetic coupling of both coils, we try to achieve that as many magnetic lines of force as possible pass through the primary and secondary windings together. The optimal magnetic coupling of the primary and secondary windings depends on a number of circumstances. For very high frequencies, higher than about 300MHz, material cores are not applicable, the coils are "air" and the magnetic coupling is given only by the tight geometric arrangement of both windings. For medium-high frequencies of 1 kHz - 300 MHz, ferrite cores (mixed iron oxides with nickel, zinc or manganese, formed by pressing) are used, which has a high resistivity, which reduces eddy current losses. For low frequencies of tens and hundreds of Hz, most often for a network frequency of 50 Hz, the material and design of the core are chosen according to the type of transformer. Small low-power instrument transformers have cores most often assembled from a layer of several dozen stacked sheets of about 0.5-1 mm thick made of ferromagnetic alloys of iron, nickel, cobalt, molybdenum. Permalloy alloy (20% Fe, 80% Ni) is mostly used. For power transformers, where high transformation efficiency and low energy losses are required, special silicon steel or amorphous metals formed by rapid cooling of molten alloys of iron, nickel, cobalt and other metals are used for the transformer core sheets. These materials have high magnetic permeability and low hysteresis and eddy current losses.
  The alternating current I₁ passing through the primary winding creates an alternating magnetic flux F₁ = N₁.I₁.µ.S/l, which is guided to the secondary winding by magnetic coupling, somewhat weakened to F₂. In the secondary coil, this alternating magnetic flux F₂, according to Faraday's law, electromagnetically induces an electric voltage U₂(t). = N₂.dF₂/dt. In the case of an ideal transformer, where the magnetic flux is identical for both windings (F₁=F₂) and there are no ohmic losses in the winding or hysteresis losses in the ferromagnetic core material, U₁.I₁ = U₂.I₂ and the transformer conversion coefficient K = U₁/U₂ = I₂/I₁ = N₁/N₂ is given by the ratio of the number of turns N₁ in the primary and N₂ in the secondary winding. When the secondary winding has fewer turns than the primary, there is a downward transformation to a lower voltage (step-down transformer), and when the number of turns in the secondary is higher than in the primary, there is an upward transformation (step-up transformer).
  There are also a larger number of secondary windings in transformers with different numbers of turns, to obtain more different voltages for individual parts of more complex circuits (e.g. primary at 220 V and secondaries at 6, 12, 24, 120 V, ...). Sometimes so-called autotransformers are also used, in which a common winding with taps for different voltages is used for the primary and secondary. In electronic laboratories, variable autotransformers are sometimes used, where a rotating contact is set along the circumference of the toroidal winding, which can sense a continuously adjustable voltage from different turns (in certain small steps depending on the number of turns). 
  
      --------------------- Small instrument transformers ----------------------------------                     ---------- Large power transformers --------
  Carefully designed transformers are very energy efficient and have ohmic and ferromagnetic losses often less than 1%. Small low-power transformers usually heat up only slightly and their cooling by the ambient air is sufficient, small fans are installed in some apparus. However, in power transformers in the energy sector, which transform powers of the order of megawatts, Joule heating of many kilowatts can occur. Therefore, there is a need to ensure their effective cooling. They are encapsulated in large metal containers with cooling "transformer" oil, which in addition to cooling also ensures better insulation properties of individual windings against high-voltage electrical discharges that would occur in the air. The oil is led through an external cooling system with fans and, after cooling, back to the transformer (pictured on the right).
=» Light sources that, when an electric current passes through them, convert part of the electrical energy into electromagnetic radiation of the optical spectrum - into light. Classic light sources - light bulbs consist of a thin metal wire, usually made of tungsten (often wound into a spiral) placed in an evacuated bulb, which is heated to a high temperature of about 1500-2000 ^oC by the passage of electric current, which leads to the thermal emission of light. Newer light sources are semiconductor LEDs.
=» Diodes and transistors (previously used vacuum tubes) are semiconductor components containing P-N junctions between P-type and N-type semiconductors. In diodes, this junction causes one-direct conduction, they function as rectifiers. In transistors, where there are 3 electrodes with P-N junctions, collector, base and emitter, they can, among other things, function as amplifiers of an electrical signal. A weak signal applied to the base can cause a much larger change in current between the emitter and collector.
=» Electric motors that convert electrical energy into mechanical energy, usually rotational. They have two coil systems: a stationary stator and a rotating rotor. In AC motors, specially wound stator coils create a rotating magnetic field, which, through another induced field in the rotor, creates a force that causes the rotor to rotate mechanically. DC motors either have a rotor made of permanent magnets, or the rotor contains slots with a commutator to which the winding is connected.
=» Switches or on/off switches that allow either turning on and off an electrical circuit, or switching the current between different parts of a more complex circuit (they are shown on the right in the figure above).
=» Circuit breakers, fuses that permanently disconnect the circuit when the current reaches an unacceptable value, thus protecting its components from damage ("burning" e.g. in the event of a short circuit). Fuses can be fusible (melting of a thin wire and interruption of the circuit), or electromagnetic circuit breakers with a coil, whose magnetic field mechanically disconnects the electrical contact by its force action.
  In addition to these individual "discrete" components, newer electrical circuits can also contain so-called integrated circuits, also called microchips, which contain a larger number of individual elements, mostly semiconductor, sometimes many thousands, in a common housing. Microprocessors and memory circuits have the highest integration density.

The most common components in electrical circuits

Electrical circuits
Note :
For the basic electronic components of resistors, capacitors and coils, when analyzing the properties of electrical circuits, we will assume that these are ideal components - resistors with resistance R, capacitors with capacitance C and coils with inductance L.
  In order to use the properties of these electronic components, we must connect them conductively to each other in an electrical circuit so that an electric current can pass through them. If the conductive path is not nterrupted in the electrical circuit, it is a closed electrical circuit. If this conductive path is interrupted at some point, it is an open electrical circuit. The simplest method here is an electro-mechanical switch whose contacts can be turned on - connected or turned off - disconnected. More complex electronic methods of closing and opening electrical circuits and their parts are also used.
  The basic parameter of an electrical circuit (and of each electrical component) is its volt-ampere characteristic - the dependence of the current I on the supply voltage U. In the simplest situations in practice, this dependence is linear according to Ohm's law I = U/R, where R is the total resistance that the individual components in the circuit offer to the electric current. This is the resistance that a conductor offers to flowing electrons. The unit of electrical resistance is 1 Ohm [?]. A current of 1 A flows through a circuit or element with a resistance of 1? at a voltage of 1 V. In a series connection, the values ??of the individual resistances are simply arithmetically added, in a parallel connection, their inverse values are added. For semiconductor components - diodes, transistors - Ohm's law does not apply exactly, the volt-ampere characteristic is more complex, nonlinear. Common metal conductors, such as copper wire, have a very small resistance of the order of milli?/meter, so it is neglected in practical electronics. Superconducting materials actually have almost zero resistance (see "....." in §,,,,,,....). Non-conducting substances - dielectrics - do not contain free charge carriers and, on the contrary, have an infinite specific resistance.
  The situation is more complicated in electric circuits with alternating current, usually with a harmonic sinusoidal waveform in time t with frequency f : I = I₀.sin (2p.f.t). Ordinary resistors behave almost the same as in direct current, their ohmic resistance does not depend on frequency. However, capacitors and coils behave completely differently. In a DC circuit, a parallel-connected capacitor can be charged once. If it is connected in series, it also charges once, but its dielectric gap is non-conductive and the circuit behaves as if it were disconnected, no current flows. In a DC circuit, on the contrary, an electric coil behaves like a conductor. However, with alternating voltage, the capacitor electrodes alternately charge, discharge and then charge with opposite polarity, which makes the capacitors effectively conductive for alternating current (the insulation gap of the capacitor is overcome by the so-called Maxwell displacement current, discussed below "..."). The effective resistance of the capacitor (capacitance) to alternating voltage is indirectly proportional to the capacity C of the capacitor and indirectly proportional to the frequency f : X_C = ¹/_{2p f . C} . In an electric coil, the alternating current inside creates a variable magnetic field, which in turn induces an electric voltage that opposes the supply voltage - self-induction occurs. The effective resistance of the induction coil (inductance) to alternating voltage is directly proportional to the inductance L of the coil and directly proportional to the frequency f : X_L =L.2p.f .
  The effective resistance of the capacitor X_C and the inductance X_L to the alternating voltage is called impedance (lat. impedire = to hinder, to be in the way, to hold back, to hinder, usually marked with X). Also "reactive" resistance or "reactance", while the resistance of the resistor R is called "active resistance". Ohmic resistances + capacitance and inductance are then added to the resulting impedance of the circuit Z. When quantifying impedances, the circular frequency w = 2p f is more often used rather than the frequency f. The capacitive impedance X_C = ¹/_{w . C} , the inductive impedance X_L = w . L and the active resistance R of the resistor are not added together arithmetically (as is the case with a resistor) in an electrical circuit, but "geometrically". The impedance of a series connection of a resistor R with a capacitor C is Z = Ö[R² - ¹/_(w.C)2], for a parallel connection of a resistor with a capacitor the resulting impedance is Z = ^{Ö(R2+w2C2R4)}/_(w2 _C.R2 ₊₁₎ .
  When alternating current passes through a resistor, the sinusoids of voltage U=U₀.sin (w.t) and current I=I₀.sin (w.t) are in phase with each other, the voltage and current reach their minimum and maximum at the same time instants. However, when alternating current passes through capacitors or inductors, a phase shift j occurs between the voltage and current - the voltage and current reach their maximum or minimum at different times. If we assign an angle of 360° to one whole period, then on the capacitor the voltage lags behind the current by 90° - this is caused by the process of alternating charging of the capacitor. On the other hand, on the inductor the voltage leads the current by 90° - due to self-induction.

Phase relationship of alternating voltage U and current I for a resistor, capacitor and coil. | Phase diagram of impedance of resistor R and reactance X

If resistors, capacitors and coils are connected in a circuit, the total phase shift between voltage and current will vary depending on the ratio of the values of R, C and L of these components. In a series-connected RLC circuit of "ideal" components of resistor, capacitor and coil, 3 significant cases can occur :
1. If X_L < X_C , the voltage on the inductor will be less than that on the capacitor and the RLC circuit will have a capacitive character - the voltage will lag the current by 0<j<-90°.
2. If X_L > X_C , the voltage on the coil will be greater than that on the capacitor and the RLC circuit will have an inductive character - the voltage will lead the current by 0<j<90°.
3. In the special case X_L = X_C, the coil will have the same voltage as the capacitor and the RLC circuit will have only a resistive character, the total phase shift will be j = 0. This special state occurs at a very specific resonant frequency f = ¹/_2p.Ö(L.C) .
The total effective impedance of the circuit is Z = Ö[R² + (w . L)² - ¹/_(w.C)2] and the phase shift between the total voltage U and the current I is j = arccos ^R/_Ö[R2_+(XL-XC)2_] .
  During the rotational movement of a point around a circle of radius r, its horizontal and vertical coordinates x,y take on the values x = r.cos j and y = r.sin j , where j is the angle between the line connecting the origin of the coordinates (0,0) and the position (x,y) of the point on the circle. Therefore, the alternating voltage/current U/I = U₀/I₀.sin 2pf.t is often represented by a circularly rotating vector U/I (0,U₀/I₀) of length |U₀/I₀| rotating with angular velocity j(t) = 2pf.t = w.t. This vector is sometimes called a "phasor", since its rotation j gives the instantaneous phase of the alternating voltage.
  Impedance is sometimes expressed in a complex (imaginary) formalism. The complex expression of impedance in algebraic form is Z = R + i.X, where R is the "effective" resistance, X is the impedance, and "i" is an imaginary unit. These two numbers R and X can be plotted graphically as a point (R,X) in the two-dimensional plane of complex numbers, where the horizontal axis has the real coordinates of R and the vertical axis has the imaginary coordinates of X. Each complex number can then be represented by a vector in this plane, starting at the origin (0,0) and ending at the point (R,X). This vector can also be expressed by its length |Z| = Ö(R² + X²) and the value of the angle j = arctang(X/R) that it makes with the horizontal axis. It is therefore a complex-expressed "phasor". This results in a phase diagram showing the complex impedance Z plotted as a vector in the complex plane, which has the real component of the impedance as the horizontal coordinate and the imaginary component of the impedance as the vertical coordinate. The impedance can then be expressed in the trigonometric form of the complex number Z = |Z| . (cos j + i . sin j), which is sometimes written in the exponential form Z = |Z| . e ^{i . j}.
  This formalism has the advantage that the same resulting relations apply to the "addition" of impedances as to the addition of DC resistances. However, a certain disadvantage is less intuitive clarity, since imaginary numbers are only model and artificial, they do not occur in real nature. The complex formalism for impedance is used more by electronics experts in the design and analysis of more complex RLC circuits.

Favorized sinusoids !
The functional course of the time dependence of electromagnetic signals can be very different in principle. However, when we observe electrical signals in various circuits, alternating voltage, radiation of electromagnetic waves and their reception, we observe in the vast majority a harmonic sinusoidal course, exact or at least approximate. It may be interesting to discuss what favors sine waves over other mathematical functions..?..
  From a mathematical point of view, the sine or cosine function has a "gift of special resisllience": when we differentiate it d sin(x)/dx = cos(x) we get cosine, which is again a sine with a phase shift of 90°. Even after integrating nsin(x)dx = -cos(x) it is just a negative cosine. Multiplying the sine wave by a constant will again make it a sine wave. The projection of a circular motion of radius r into the coordinates x and y oscillates harmonically as x(t) = r . cos w.t , y(t) = r . sin w.t , where w is the angular frequency.
  All oscillatory movements caused by a force F which is proportional to the deviation x from the equilibrium state F = -k.x - the movement of a classical pendulum, waves on the water surface, elastic oscillations of particles in a material environment, electrical oscillations in an LC oscillator - occur with a resulting deflection of the form x(t) = r . sin w.t . From an electronic point of view, a sinusoidal signal is the only shape that does not change its character when it passes through an electrical circuit containing capacitances, inductances and resistors. And each configuration of an electrical signal or electromagnetic wave can be decomposed using Fourier analysis into a superposition of a smaller or larger number of harmonic sinusoidal signals or waves of different frequencies and amplitudes..!.. Any nonlinearity in an electrical circuit distorts the pure sinusoidal waveform, which is manifested by the appearance of signals of the so-called higher harmonics, which again behave as sinusoids of different frequencies.
  Sine oscillations and waves are naturally produced by nature in the field of mechanics and electrodynamics; similarly to the field of gravity in the universe, elliptical paths of movement of planets around stars, orbits of moons around planets, or mutual orbits of stars in binary and multiple stellar systems arise naturally. Sinusoids and cosinusoids are therefore natural functions that can be used to model and quantify a number of processes in nature using simple harmonic oscillators.
  The vast majority of electrical energy for the world economy and consumption in our homes is produced in alternators, where a rotating magnetic field induces an alternating current in coils of exactly sinusoidal form with frequency of 50 or 60 Hz.
How fast is electricity ?
In terms of speed, we encounter two extremes in electricity: the speed of propagation of the electromagnetic field and the speed of movement of electrons in conductors. It is a notorious experience that when we turn on the switch, light bulbs many meters away (even kilometers - city lighting) immediately light up. Or a telephone connection even over long distances is immediately established (we do not consider complex relay connections here). So one could conclude from this that the electrons are moving at a high speed in the conductor. This conclusion would be completely wrong.
  Although the electrons in the conductor, even without the electrical circuit switched on, move at room temperature at very high speeds of the order of thousands of km/s., however, these are only microscopic completely chaotic thermal movements that in total do not create any electric current. When we apply a voltage to the conductor, in addition to their chaotic movement, they begin to move slowly in one direction, towards the positive voltage - the so-called drift movement. However, the speed of this movement is very small, on the order of millimeters/second. So how come the remote lightbulb lights up immediately? When the switch is turned on, the electrons almost immediately begin to move along the entire length of the connecting wire, and the bulb lights up immediately. That practically instantaneous effect is caused by the speed of propagation of the electromagnetic field along the conductor, which is close to the speed of light (see below "Speed of propagation of an electromagnetic signal"). So "snails activated at the speed of light"....
  The movement of electric charges in space and time is generally described in field theory using the current density j(x, y, z, t) º r . v , where v is the instantaneous velocity of charges at that point (x, y, z); electric current flowing through a given surface S then I = _S òò j dS. The law of conservation of electric charge then states that the change in charge contained in each given spatial area V must be equal to the amount of charge that passes through the closed surface S = ¶V surrounding this area :

(1.31a)

Using Gauss's theorem, the well-known equation of continuity follows

expressing the law of conservation of electric charge in differential form.

Coulomb's law of excitation of the electric field by charges
The fundamental law of electricity is Coulomb's law of excitation of an electric field by electric charges (in the previous §1.4 we stated it under the number (1.20b)) :

which expresses the mutual force action of two (point) electric charges q₁ and q₂ placed in a vacuum at a distance r from each other (r° is the unit position vector of both charges relative to each other). The "-" sign expresses the fact that charges of the same name (same polarity) repel each other. The value of the constant k depends on the system of units used. In fundamental physics, k=1 is declared (which naturally defines the unit of electric charge by its force action per unit distance *), in the SI system k = ~8.988×10⁹ N m² C^-2 and the unit of electric charge is 1 Coulomb (C).
*) Unfortunately, the historical development of physics has led to the fact that in the SI system of units, charge is not primarily quantified by its electric force effects, but only indirectly by the magnetic effects of electric current (the unit of current is Ampere; one Coulomb is then defined as 1A/1s).
  In the SI system, Coulomb's law is written in the form with coefficient k = 1/4pe₀ :

where e₀ is the permittivity of vacuum e₀ = ~8.854×10^-12 F.m^-1. The permittivity of material media will be discussed below - "Coulomb's law in material media".
 Etymology: Lat. permittere = to forward, to allow - to what extent the material media allows electric forces to penetrate.
  For the action of force in space "at a distance", physics introduces the concept of a physical field, which is a space in which forces of a given type act on (test) particles. In electricity, it is an electric field excited by electric charges (and also by electromagnetic induction). If the electric charges do not move, it is an electrostatic field that is quantified by the electric intensity vector E_el, which is the force acting on a unit test charge q, i.e.

The electric force and intensity E_e are generally a function of the location - coordinates in the studied space. For the sake of brevity of notation, instead of the individual coordinates x,y,z, we will use the position vector r (radius vector) - the line connecting the origin of the coordinate system and the investigated point, where, for example, the charge is located or where we determine the intensities E(r) and potentials j(r) of the fields.
  In addition to the intensity E_el , the electric potential j is also introduced in the electric field. It is a scalar quantity describing the potential energy of an electric charge in an electric field - the amount of work required to transfer a unit electric charge from a reference (default) point, where the potential is considered to be zero, to a given point r in the electric field. The reference point with zero potential is usually taken to be a point infinitely distant from the system of charges, where no electric field is acting (at least in the limit); in practice, the surface of the Earth (grounding) is taken. The potential j of an electric field is related to its intensity E by the relation

where grad f = [¶f/¶x, ¶f/¶y, ¶f/¶z]  is a vector differential operator quantifying the "steepness of the slope" - gradient - of the scalar field f in the direction of the coordinates x,y,z.
  The potential difference of two points gives the electric voltage U between these two points, the unit of which is 1 volt [V]. A voltage of 1 Volt is such that, in order to overcome it by a point charge of 1 Coulomb, it is necessary to perform (or release - depending on the polarity) work of 1 Joule. The voltage U between two points r₁ and r₂ in an electric field of intensity E(r) is given by the difference

(integrated along the line "l" between the two points). In practice, the electric voltage is quantified not so much for different points in space, but between the electrodes to which it is supplied from a certain source. When a charge q is moved between points with a voltage difference U, work is performed (or released) W = q . U.
   Coulomb's law can then be expressed in terms of the electric field intensity E_el (we will omit the index "el" in the following) excited in space around a point electric charge Q :

The excited electric potential here is then

The potential depends only on the distance |r|, not on the direction relative to the exciting charge Q.
 In practice, the electric field is usually excited not by randomly distributed electric charges, but by electrodes to which an electric voltage [V] is applied from a suitable source, which can be a galvanic cell, an electro-mechanical generator, an electronic circuit, or other device or material configurations.
 Electric field in a material environment
Coulomb's law in the form (1.20b) also applies not only in vacuum, but in an electrically homogeneous and isotropic material medium called a dielectric

while the proportionality constant k expressed in the form k = 1/4pe, where e is the permittivity (dielectric constant) of the given material medium. The permittivity of a vacuum is e₀ = ~8,854×10^-12 F . m^-1 .
  The names "insulator" and "dielectric" are sometimes terminologically distinguished (a dielectric is an insulator in which particles are polarized). Due to the atomic structure of all known substances, polarization of atoms and molecules always occurs, so from a physical point of view the terminological difference is irelevant.
  There are basically two types of dielectrics. Either the substance is composed of polar atoms or molecules - permanent dipoles, which rotate in the direction of the field under the influence of an external electric field. Or the substance is composed of originally non-polar particles, which are, however, polarizable under the influence of an external field. In both cases, polarization occurs when placed in an electric field, with the polarized dipoles acting against the external field and polarization reducing the resulting intensity E of the electric field in the dielectric compared to the field in a vacuum.
  The way in which the electrical polarization and magnetization of atoms and molecules of the material environment arise and how it is reflected in the intensities of the resulting electric and magnetic field is clearly shown in §1.1, passage "Electromagnetic and Optical Properties of Substances" monograph "Nuclear Physics and Ionizing Radiation Physics".
  The permittivity of materials e is often quantified using the relative permittivity e_r = e/e₀, also called the dielectric constant. It indicates how many times the electric force between charges decreases when they are placed in a given medium instead of a vacuum (at the same time it indicates how many times the capacitance of a capacitor increases when a dielectric is inserted between the plate electrodes). For vacuum, of course, e_r = 1, for air and other dilute gases it is also close to 1. For wood and pressed paper, e_r is ~2-2.5, for plexiglass about 3.5, for water ice 4.8, for diamond 5.5, for water e_r =80 (it is a polar compound).
  For our theoretical analysis of the nature of electrical phenomena, we do not need to introduce the quantity D =e . E called electrical induction. We will only use it below for the formulation of Maxwell's equations in a material environment (1.38´-41´).
  In a vacuum, the dependence between the size of electric charges and the excited electric field is exactly linear (direct proportionality) with a coefficient of 1/4pe₀ up to colossally high intensities of about 10¹² Volt/micrometer. It is limited to the quantum level (brief discussion "What is the strongest electric field?"). In a dielectric material environment under normal situations, this linearity is also preserved, only with a somewhat lower coefficient of 1/4pe. Linearity here can be disturbed only for very strong electric fields, when the phenomenon of the dielectric's electrical strength can manifest itself :
  At a high value of the electric field intensity, the insulating properties of the dielectric can be violated - an electrical breakdown and an avalanche-like passage of a large number of charged particles (mostly electrons) can occur, and a spark can jump between the electrodes of opposite polarity. Under the influence of a strong electric field, the originally bound electrons are released and can accelerate so much that in collisions with neutral atoms and molecules, more and more electrons are ejected, which creates an avalanche-like current, an electrical breakdown, within a few nanoseconds. If the electrodes are powered by a "harder" electrical source of greater power, a more permanent discharge can occur at the breakdown point - an electric arc, with thermal effects of melting or igniting the material. The value of the breakdown voltage [kV/mm] depends primarily on the type of insulating (dielectric) material, but also on the configuration of the electrodes, on the possible content of impurities, microscopic dislocations and free electrons or ions, which are also contained in trace amounts in insulators. For air it is about 2-3 kV/mm, for glass and porcelain about 10-30, for PVC 30-50, polyethylene about 100, for polyester up to 180 kV/mm.
  In addition to classical dielectrics, which are polarized by an external electric field and disappear after the field is removed, there are a few rare materials that can be permanently polarized even after the external electric field is removed - so-called electrets, electrical analogs of permanent magnets. The basic method of creating an electret consists of three steps: 1. Melting a suitable dielectric substance, e.g. paraffin or resin. 2 Inserting the molten substance into a strong electric field - between electrodes to which a high voltage of several kilovolts/cm is applied. Here, the atoms or molecules inside the molten substance are polarized. 3. Allowing the molten substance to cool and solidify in this electric field. The polarized molecules in the solidified substance lose mobility ("freeze") and retain their polarization even after the electric field is turned off. An electrostatic field will then be permanently active around the electret. Weak electrets also occur naturally in nature, mainly in various forms of silicon oxide. For artificially produced electrets, some easily meltable dielectric materials are suitable, such as paraffin (wax), resins, polymerized plastics such as fluoropolymers, polypropylene, PET, PTFE, and sulfur, ...... In addition to the thermal method, electrets are more recently prepared by corona discharge, irradiation of thin layers with soft X-rays, injection of electrons using an accelerator. Electrets are used electromechanically in small electret microphones, in sensor transducers in movement and deformation monitors, air filtration media, xerography, memory devices, integral detectors of ionizing radiation (especially when measuring radon concentration)...
    From the point of view of field theory, Coulomb's law can be expressed in the form of Gauss's theorem of electrostatics (Fig. 1.3a)

(1.32a)

from which the differential equation follows

Fig.1.3. Excitation of electric and magnetic fields by electric charges and currents.
a ) The total electric charge Q contained in the space inside any closed surface S is given by the Gauss theorem flux of the electric field E over the closed surface S .
b ) Circulation of the vector magnetic field B around the closed curve C is proportional to the total electric current I flowing through the surface S bounded by the curve C .
c ) The electromagnetic field excited by a system of moving electric charges is given by the distribution of charges and currents, retarded always by the time required by the field to overcome the distance r - r' from the individual places dV' of the system to the examined place r .

In nature and in electronic applications may arise even strong electric fields at a volage of several millions of volts. For interest, we can give a small discussion, what is the strongest electric field that can be achieved? :
What is the strongest electric field can be ?  
In classical (non-quantum) physics, the electric field in a vacuum can be arbitrarily strong, almost to infinity (in a material environment, however, this is limited by the electrical strength of the dielectric) . From the point of view of quantum electrodynamics , however, even in vacuum there is a fundamental limitation caused by the existence of mutual antiparticles of electron and positron : it is not possible to create an electric field with an intensity stronger than E _e-_e+ = m _e² c³ / e.h = 1.32 × 10¹⁶ V/cm, where m_e is the rest mass of the electron or positron. When this intensity is exceeded, the potential gradient is higher than the threshold energy 2m_e c² and a pair of electrons and positrons is formed, which automatically reduces the intensity of the electric field. Such a strong electric field has not yet been created, with conventional electronics this is not possible; strong impulses from extremely powerful lasers could be a certain possibility in the future ...
  At the end of §1.6, in the passage "Nonlinear electrodynamics", a purely hypotetical model of classical relativistic nonlinear electrodynamics will be discussed.

Magnetic field excitation
In addition to the electric forces acting even between stationary charges, there are also magnetic forces acting only between moving charges in the field of electricity. The space in which these forces act is called a magnetic field. If an electric charge q moves in this space with a velocity v, it is acted upon by a force

where B is the magnetic field intensity (for historical reasons called magnetic induction), "x" means the vector product. It is called the Lorentz force. This force is perpendicular to the direction of the velocity v of the particle. The SI unit of magnetic induction B is 1 Tesla: In a homogeneous magnetic field of intensity (induction) of 1 Tesla, a linear conductor of length 1 meter, stretched perpendicular to the magnetic field lines and flowing with a constant current of 1 Ampere, exerts a force of 1 Newton. In the CGS system, the unit of magnetic induction is 1 Gauss = 10^-4 Tesla.
  The magnetic field is excited by moving electric charges, ie electric current, according to the Biot-Savart-Laplace law. The basic form for a point electric charge Q, moving with a velocity v, gives how strong the magnetic field B will be at a distance r - at a location with a position vector r = r . r⁰ :

The Biot-Savart law is usually formulated in differential form for an electric current flowing through a linear conductor :

where dl is an element of the length of the conductor through which a stationary electric current I flows, r is the distance of the measured location, and r° is the unit position vector pointing from this current element to the investigated location ("x" is the vector product). The value of the constant k depends on the system of units used. In fundamental physics, k=1 is declared, in the SI system it is k= m₀/4p, where the coefficient m₀ is the permeability of vacuum, in SI units it is m₀ = ~1,257×10^-6 N . A^-1 :

Etymology: Lat. permeare = to pass through, to let through - here the property of a substance to let through or amplify a magnetic field.
  The permittivity e₀ and permeability m₀ are related to the speed of light c in vacuum by the relation c = 1/Ö(e₀.m₀), as will be shown in the section "Electromagnetic waves". From the comparison of relations (1.30b) with (1.33a,b) we see that magnetism is inextricably linked to the dynamics of the movement of electric charges: The magnetic field exerts a force on moving charges and is also created by the movement of charges. From relation (1.34a,b) it is then seen that a magnetic field can also be created by a time-varying electric field.
 Magnetic field in a material environment
The Biot-Savart-Laplace law of excitation of a magnetic field in the form (1.33a,b) is valid not only in a vacuum, but also in a homogeneous and isotropic material environment, where the permeability of the vacuum m₀ is replaced by the magnetic permeability of the given material m. Most substances (gases, water, dia- and paramagnetic metals such as copper, aluminum, gold, .... ) have permeability almost equal to m₀; only the ferromagnetic metals iron, cobalt, nickel and their alloys have permeability on the order of 100× to 100000 times higher.
  For materials, the relative permeability m_r = m / m₀ compared to vacuum is introduced, according to which materials are divided into 3 categories :
-> Diamagnetic m_r < 1, which slightly weaken the magnetic field. The inner shells in the atomic shell are fully occupied by electrons, the orbital and spin moments are completely compensated, the resulting magnetic moments of the atoms are zero. When a magnetic field is excited, a weak magnetic moment is induced in the material, acting against the external magnetic field. These are some metals such as copper, silver, gold, mercury, most organic substances, water, inert gases.
-> Paramagnetic m_r > 1, which slightly strengthen the magnetic field. The inner shells in the atomic shell are not fully occupied by electrons, the magnetic moments of the atoms are not zero, but are irregularly oriented in all directions. After the external magnetic field is excited, the moments of the atoms are oriented in the direction of the external field, which slightly strengthens the resulting magnetic field. These are, for example: metals aluminum, magnesium, chromium, platinum ...
-> Ferromagnetic m_r >> 1, which significantly enhance the magnetic field. This is due to the fact that these substances contain spontaneously arising magnetic domains, which have their magnetic moments oriented identically. Without the presence of an external magnetic field, the directions of the individual domains are chaotically oriented, so the resulting magnetic moment is zero. However, it is enough to excite a relatively weak magnetic field for the magnetic moment of the domains to orient in the direction of the external field, which leads to a significant amplification of the external magnetic field. These are mainly iron, cobalt, nickel, gadolinium, ruthenium and their alloys (such as permalloy 20% Fe, 80% Ni - m_r~100,000; suprrmalloy 20% Fe, 75% Ni, 5% Mo - m_r~800,000). Ferromagnetic properties only work at sufficiently low temperatures. At temperatures higher than the so-called Curie point, the magnetic domains disappear and the material changes from a ferromagnetic to a paramagnetic state. For iron, the Curie temperature is 1040 ^oC, for neodymium magnets only about 70-120 ^oC.
  However, in very strong external magnetic fields, almost all domains are already oriented in the direction of the field and further increase in its intensity leads only to a milder paramagnetic enhancement. The ferromagnetic enhancement of the external field is therefore dependent on the intensity of the external field, showing a saturation state.

Left: Magnetization dependences for basic categories of substances. Right: Hysteresis loop of ferromagnetic materials.

Magnetization characteristics - the dependence of the resulting magnetic induction on the excitation intensity - are linear straight lines for dia- and paramagnetic materials, because theirm_r is constant. For ferromagnetic materials, it is no longer a straight line, because their m_r changes during magnetization depending on the intensity of the magnetic field. A relatively complex magnetization characteristic is manifested here, showing a saturation state B_S .
  In addition, a number of ferromagnetic materials exhibit a special dynamic "memory" behavior called hysteresis - irreversibility of magnetization processes. If we excite a sufficiently strong magnetization, when the excited magnetic field is reduced again, the resulting magnetization curve does not return to the same functional dependence as before when increasing, but has somewhat higher values than before. When the intensity of the magnetic excitation drops to zero, a certain non-zero remanent magnetic induction B_r remains, which the material retains (approximately 0.1-1 T). Part of the magnetic domains remains oriented. To cancel the remanent magnetization, it is necessary to apply a certain coercive magnetization of the opposite direction, the further increase of which then leads to a magnetization curve of the opposite polarity, also showing the saturation state -B_S. When alternating magnetization in opposite polarities, a closed curve is created, called a hysteresis loop.
  According to the width of the hysteresis loop, ferromagnetic materials are divided into two categories :
--» Magnetically soft have a narrow hysteresis curve with a low B_r value, so that after the external magnetic field is removed, their own magnetic field also practically disappears. This property is possessed by pure iron (with a low carbon content <0.1%), permalloy.... .. ......
--» Magnetically hard with a wide hysteresis loop and a large B_r value. After the external magnetic field is removed, they remain magnetized and behave like a permanent magnet. Their magnetic field can be removed by an external magnetic field of opposite polarity. This is how iron-steel with a high carbon content, tungsten steel, chromium and cobalt behave. The strongest permanent magnets are prepared from the rare earth elements samarium (Sm-Co-5, Sm-2-Co-17) and neodymium (Fe-Nd-B). They achieve high remanence values B_r ~ 0.8-1.4 Tesla.
  For our theoretical analysis of the nature of magnetic phenomena, we do not need to introduce the quantity H =e . B called magnetic intensity. We will only use it below for the formulation of Maxwell's equations in a material environment (1.38´-41´).
  From the Biot-Savart's law (1.33b) follows the Ampere's law

(1.33b)

according to which the curve integral - circulation - of the magnetic induction vector along any closed curve C is proportional to the total current flowing through the surface S , which this curve surrounds (Fig.1.3b).

The integral on the left side of Ampere's law depends only on the curve C = ¶S, so for equation (1.33b) to be generally valid, the area integral on the right side must be the same for all surfaces S having the curve C behind the contour. Using Gauss sentences can be easily shown that this is only fulfilled if div j = 0, i.e. when it is the stationary electric current that does not cause changes in the distribution of electric charge in the vicinity of the curve C. For general non-stationary currents, it is therefore necessary to generalize Equation (1.33b) to be compatible with the continuity equation. Substituting in the equation of continuity (1.31b), which also applies to non-stationary currents, for r from equation (1.32b), we get

div [ j + (1/4p) ¶E / ¶t] = 0 .   

This vector is found, j + ¹/_{4 p} ¶E / ¶T whose divergence is always equal to zero, and that in the stationary case, coincides with a normal density "conductive" stream j. The expression j_Maxw = (1/4p) ¶E/¶t is called the Maxwell displacement current and can exist even in a vacuum without the presence of real electric charges. Maxwell proposed in the case of a non-stationary field in equation (1.33b) to replace the current density j with the vector j + (1/4p) ¶E/¶t , or stated the hypothesis that the displacement current has the same magnetic effects as the normal "conductive" current of real electric charges :

(1.34a)

The magnetic field is thus excited by the total effective current

This Maxwell's hypothesis proved to be very correct and fully corresponds to all experiences with electromagnetic phenomena. The Maxwell displacement current is, for example, that current which "overcomes" the insulating layer of capacitors and causes their "conductivity" for alternating currents. If we have a planar capacitor with a plate surface S , then between the intensity of the homogeneous electric field in the gap and the charge of the capacitor q, the relation E = 4p q/S holds, so that the instantaneous current flowing through the capacitor I = ¶q/¶t = S. (1 /4p) ¶E/¶ t = S. j_Maxw is given by Maxwell current.

The displacement current, which - although not formed by the motion of real electric charges - has normal magnetic effects, finds its analogy in the gravitational field, where even in a vacuum without real material bodies there is effective Isaacson energy and momentum of gravitational wave, which has gravitational effects (curves spacetime) like any other matter (see §2.7-2.8).

By converting the integral along the curve C using the Stokes theorem to the integral over the surface S surrounded by this curve, we obtain the equation of excitation of the magnetic field by an electric current (conductive and displacement) in a differential form

From this equation it is clear that the magnetic field can be created not only by the movement (current) of electric charges, but also by a time-varying electric field .
 Note: Note the coefficient 1/c = 3.33×10^-9 s/m in all relations (1.33)-(1.34) between the magnetic field intensity B, the current I, j and the electric field E. The very low value of this coefficient reflects the fact that the magnetic field arising from the motion of charges or electromagnetic induction is generally very weak compared to the electric field (excited by the same charges). For electromagnetic waves, this fact is briefly discussed below in the passage "Lesser-known properties of electromagnetic waves".
  A strong magnetic field of tens of Tesla can be created in the laboratory by passing a strong current of several thousand Amperes through electromagnetic coils, mostly superconducting (§1.5, section "Electromagnets in accelerators" and §1.3, section "Tokamak"). In the universe, powerful magnetic fields arise during the gravitational contraction and collapse of compact stellar objects such as white dwarfs and neutron stars, when the acceleration of rotation under the law of conservation of angular momentum enormously thickens the magnetic field lines and the magnetic field can increase to enormous values of up to ~10⁸-10¹² Tesla (§4.2, section "Astrophysical significance of supernovae").

Another basic law of electromagnetism is the finding that magnetic field lines are continuous and closed curves. In other words, the magnetic field is source-free, there are no magnetic "charges" (monopolies) *) from which magnetic field lines originate or enter (unlike electric charges, where electric field lines begin and end). Therefore, exactly as many magnetic field lines must enter from the closed surface S as enter them, ie the magnetic flux from the closed surface is equal to zero :

(1.36a)

By converting the area integral to the volume integral using the Gauss theorem, we get the equation

which is a mathematical expression of the principle of continuity of magnetic field lines in differential form.
*) We leave aside here the Dirac's hypothesis about of magnetic monopoles, based on the formal requirement of symmetry of the equations of electrodynamics. Experiments trying to find magnetic monopoles have not yet yielded any results. However, magnetic monopoles are considered in quantum unitary field theories and their possible significance for the cosmology of the very early universe is also discussed (§5.5 "Microphysics and Cosmology. The Inflationary Universe.").

Electromagnetic induction
The generation of an electric field by a time-varying magnetic field is expressed by Faraday's law of electromagnetic induction

(1.37a)

according to which the electromotive force (voltage) U s ?n_C E dl induced along a closed curve C is proportional to the speed with which a change in magnetic flux F s nn_SB dS by surface S enveloped by the curve C. In the integral on the right side it does not matter the choice of the surface S surrounded by the given curve C , because the magnetic field is source-free (div B = 0). By converting the curve integral on the left side using the Stokes theorem to the area integral we get the law of electromagnetic induction expressed in differential form :

Maxwell's equations
The outlined application of the mathematical apparatus of differential and integral calculus to the empirically determined laws of electromagnetism (ie to the knowledge of Coulomb, Ampere, Faraday, Biot, Savart, etc.) and their generalization was performed by J.C.Maxwell, who arrived at a complete system of basic equations of the electromagnetic field and summarized the individual findings into a comprehensive theory. J.C.Maxwell formulated his very important equations in 1865, first in the coordinate form of the components x,y,z of the electric E and magnetic B fields and the current density j. This resulted in a quite complex system of more than 20 equations (sometimes a more complex so-called tetrad formalism is also mentioned, but it was soon abandoned). They were reformulated in the compact vector form (1.38)-(1.41) by O.Heaviside in 1884.
  These Maxwell's equations (1.31b) to (1.37b), which we gradually derived above, can be clearly summarized in differential form as follows :

	(1.38)
	(1.39)
	(1.40)
	(1.41)

	(1.38)	S I
	(1.39)
	(1.40)
	(1.41)

These equations determine the electric and magnetic fields E and B excited by a given distribution of charges and currents r and j. The first pair of Maxwell's equations describes the generation of electric and magnetic fields by material sources, ie the density of electric charge r and current j protruding on the right side; the second pair expresses other internal properties of the field. It can be seen from equations (1.38) and (1.40), that the electric E and magnetic B fields can mutually generate each other with their time variability --> electrodynamics .
The Lagrangian for the electromagnetic field
We derived the Maxwell equation of the electromagnetic field here physically - inductively from Coulomb's, Biot-Savart's, Ampere's, Faraday's law. In theoretical physics, the laws of motion and field equations are often derived in a deductive manner using the variational principle of least action [165]. The classical physical system is described by the integral of the action S

where L is the so-called Lagrange function - Lagrangian, describing all dynamic characteristics q_i of a given system and their time derivatives q^._i , n is the number of degrees of freedom. The variational principle of the smallest action dS = 0 then leads to Lagrange's equations, from which the equations of motion or field equations of the studied system follow (for more details, see §2.5 in the section "Variational Derivation of Gravitational Field Equations") .
For the electromagnetic field, the Lagrangian has the form:

Maxwell's equations (1.38-41) can be derived from the variational principle of the smallest action with Lagrangian (1.42).

Electromagnetic field in the material environment - electrodynamics of the continuum
In our theoretical analysis, we consider the electromagnetic field mainly in vacuum, which is the basic natural environment for fundamental physics. For the sake of completeness, we will briefly outline here how the electromagnetic field behaves in material environments (the material "continuum"). The interaction of electric and magnetic fields with atoms and molecules of matter leads to their electric polarization and magnetization, which is reflected in the vectors of electric and magnetic field intensity. The way in which the electrical polarization and magnetization of atoms and molecules of the material environment arise and how it is reflected in the intensities of the resulting electric and magnetic field is clearly shown in §1.1, passage "Electromagnetic and Optical Properties of Substances" monograph "Nuclear Physics and Ionizing Radiation Physics".
  To quantify this influence of the material environment on electric and magnetic field intensities, two new vectors are introduced: electric field induction D and magnetic field intensity H (historically confusing terminology is discussed below in the note " Intensity < -   > induction in electromagnetism ?) . With basic electrical quantities E and B in a vacuum are related by relationships :

where e is the electrical permittivity of the substance (also called dielectric constant) describing the attenuation of the electric field due to the polarization of the substance, m is the magnetic permeability indicating the amplifying or attenuating effect of magnetization of the substance on the magnetic field.
  Maxwell's equations of the electromagnetic field in a material environment (electrically non-conductive) can then be written in the same form (1.38-41) as in a vacuum, in which, however, "vacuum" intensities E and B are replaced by "substance" vectors D and H at appropriate places :  

(1.38´-41´)

where the relationships between E and D = E / e , B and H = B / m contain the material coefficients of electrical permittivity e and magnetic permeability m. They also include possibly inhomogeneities and nonlinearities of polarization and magnetization - in some material environments and at high field intensities nonlinear electrodynamics can manifest itself (theoretical possibility of nonlinear electrodynamics even in vacuum for extremely strong electromagnetic field is discussed at the end of the following §1.6 "Four-dimensional spacetime and special theory of relativity", passage "Nonlinear electrodynamics").
  The laws of continuum electrodynamics, summarized in Maxwell's equations (1.38´-41´), describe all electromagnetic phenomena occurring in material environments (see the already mentioned passage "Electromagnetic and optical properties of substances"). Due to the interaction of electric and magnetic fields with atoms and molecules of matter, time changes in fields (see "Retarded potentials" below) and electromagnetic waves in matter environments propagate at a rate c´ lower than in vacuum: c´ = 1/Ö(e.m) < c = 1/Ö(e_o.m_o) = 2,998.10⁸m/s @ 300,000 km/s (for light this leads to known optical phenomena of refraction of light rays during the transition between substances with different "optical densities" - different refractive index caused by different velocities c´) .
Terminological note: Intensity <--> induction in electromagnetism ? 
The term intensity in science (even in ordinary expression) characterizes the degree of power, mightines, yield of an event, a phenomenon - here the power of the field. The intensity of the electric field E really expresses an electric force acting in this field on a particle with a unit electric charge (in suitable units) . The "intensity" of the magnetic field B should analogously express the magnetic Lorentz force (the second term in (1.30) ) acting in this field on a unit-charged particle, when perpendicular to the unit velocity. However, the quantity B, describing the actually acting magnetic force, is called in magnetism not intensity, but magnetic induction! And the "intensity" of the magnetic field is called the derived quantity H (= B/m) "corrected" for the magnetic permeability of the material environment. At the same time, it is in electrostaticselectric induction D (= E/e) derived quantity characterizing the electric field minus the effect of dielectric polarization. So it's the opposite ...
  This unfortunate "crossing" of the names "intensity-induction" arose during the historical development of the science of electricity and magnetism, when magnetism was explained by fluid theory, analogous to electrostatics. And unfortunately it has already remained so... In our treatises, therefore, we will often understand the vector B (conventionally called magnetic induction) by "magnetic field strength".
Note: The word "induction"here characterizes the electrical and magnetic changes in substances caused by their insertion into electromagnetic fields. Do not confuse with electromagnetic induction (1.37) ..! ..

Properties of Maxwell's equations
Let's briefly note some general properties of the system of Maxwell's equations (in vacuum). First of all, from the 1st pair of Maxwell's equations we get (by applying the operation "div" on equation (1.38), the operation "¶/¶t" on equation (1.39) and their suming) the continuity equation div j + ¶r/¶ t = 0. The distribution and movement of electric charges cannot therefore be entered completely arbitrarily; in order for Maxwell's equations to be satisfiable, the continuity equation must be satisfied. In other words, the electric charges around them excite the electric and magnetic fields, so that they preserve themselves - the continuity equation is a consequence of the field equations.
  Equations (1.39) and (1.41) do not contain time derivatives and therefore have the character of boundary conditions; the remaining two equations (1.38) and (1.40), which (using the "div" operation on both sides) can be adjusted to the form

¶/¶t (div E - 4pr) = - 4p (div j + ¶r/¶t) = 0 , (continuity equation) ,

^¶/_{¶ t} div B   = -c div rot E    º    0 ,

then they guarantee that if these initial conditions div E = 4p r and div B = 0 are satisfied at some time t = 0, they remain satisfied at all times.

Field potentials
In field theory, in addition to the intensity vectors of a given field, it is advantageous to introduce field potentials, which are quantities whose derivatives (differential forms) indicate the respective intensities. In electrostatics, the intensity of the electric field E can be expressed as a gradient of the electric potential j (E = - grad j), which identically satisfies the equation rot E = 0. In magnetism, the equation div B = 0 holds, so there must be a quantity (vector field) A, such that B = rot A. From the second pair of Maxwel equations it follows, that the vectors E and B in the case of a general electromagnetic field can be expressed by quantities j and A in the form

By introducing such an electric potential j and a magnetic vector potential A, the last two Maxwell's equations are fulfilled identically.
  Since the field intensities depend only on the derivatives of the potentials, these potentials are not determined unambiguously; different values of the potentials may correspond to the given fields E and B. E.g. to A can be added to an arbitrary constant vector, and to j arbitrary constant without changing the values of intensities E and B. In general, the magnetic field B = rot A does not change, if we add to A a gradient of any function f (rot grad f º 0); in order while doing so not changed even the electric field E (1.43a), it is also necessary to add the term -(1/c).¶f/¶t to the potential j. Therefore, if we perform this so-called calibration or gauge transformation *) of potentials

where f(r,t) is an arbitrary scalar function of place and time, the corresponding electromagnetic field does not change ( E ® E' = E , B ® B' = B ). This certain "freedom" in choosing potentials allows you to select the shape of the potential (to carry out their "calibration - gauge") so that it was possible best for the specific problem.
*) The procedure of gauge transformation in field theory is generally discussed in §B.6, passage "Calibration - gauge - transformation; calibration - gauge - field".

The retarded potentials
Maxwell's equations (1.38) and (1.39), expressed with the potentials by substituting from (1.43a, b), generally have a rather complex form

These equations can be greatly simplified, if the so-called Lorentz gauge condition is prescribed for potentials :

this condition can be satisfied by the transformation (1.44) with the function f satisfying the equation Df - (1/c²).¶²f/¶t² = div A + (1/c).¶j/¶t). During this gauge, the Maxwell's equations, expressed by means of potentials, take on a separated and symmetrical form of the d'Alembert equations

(1.46a) (1.46b)

where o º ¶²/¶x² + ¶²/¶y² + ¶²/¶z² - (1/c²)¶²/¶t² is d'Alembert's differential operator. In mathematical physics, it turns out that the general solution of these equations has the form *)

(1.47a, b)

where r = (x, y, z) is the position vector of the point at which we determine the potentials, r' = (x', y ', z') is the position vector of the volume element dV' = dx'dy'dz' dureng the integration of charge density and current, j_o and A_o describe the external field acting on the system (or integration constants). Equations (1.47a, b) show that at a given place r and at a given time t , the field is given not by the instantaneous distribution of charge and current in the whole space, but by the distribution retarded (delayed in the past) always by time |r - r '|/c , which is needed to overcome the distance R = |r - r '| at speed c from individual points (x', y', z') of the source system to the examined place (x, y, z) - see Fig.1.3c. Solution (1.47) is therefore called retarded potentials. The change (disturbance) in the electromagnetic field (caused, for example, by a change in the distribution of charges) therefore propagates at a final speed equal to the speed of light c.
*) Note: In the previous §1.4 and in the first half of this §1.5, we marked area and volume integrals by double and triple integrals: òò_S f(...) dS and òòò_V f(...) dV. In the following, however, for brevity we will use only one integration sign: ò_S f(...) dS and ò_V f(...) dV with indicating the surface S and the volume V .

Relativistic electromagnetism
In classical electrodynamics, electric and magnetic fields are separate fields, interconnected only by the laws of excitation and induction, summarized in Maxwell's equations . In the special theory of relativity (discussed in the following §1.6 "Four-imensional spacetime and special theory of relativity"), created by A. Einstein on the basis of a careful analysis of electromagnetism, however, we will see that the division of electromagnetic forces into separate electrical and magnetic ones is not fundamental, but may depend on the frame of reference. Simply put, what appears to the observer in one frame of reference with the rest distribution of electric charges as a purely electric force will appear to the moving observer in another frame of reference as a magnetic force, resp. combination of electric and magnetic forces.
 Is the magnetic field a relativistic manifestation of the electric field ?
By emphasizing the relationships between the electric and magnetic fields and the motion of charges, some authors have come to the conclusion that the magnetic field as such does not exist independently, but can be considered a relativistic manifestation of the electric field. If we have a system of static electric charges in one frame of reference, only the electric field will act here, we will not observe any magnetic field. However, a moving observer in another frame of reference, who will look at the same system of charges, will see a flow of charges, according to electrodynamics, exciting a magnetic field - a magnetic field arises associated with the motion of charges - with an electric current - according to the Biot-Savart-Laplace law. However, from the point of view of STR, the magnetic field appears here as a "relativistic product" during Lorentz transformations of coordinates in the presence of a stationary electric field. However, this can be interpreted in this way only in the simplest situations of the distribution of electric charges and during translational motion. This does not apply to rotational motions, to the magnetic fields of permanent magnets, or to induced magnetic fields. However, all these connections are based on the mutual relationship between the electric and magnetic fields, expressed in Maxwell's equations. Objectively, we have an electric field producing a force independent of the speed of the charges, and a magnetic field producing a force depending on the speed of the charges, which coexist and influence each other, and both are necessary for electromagnetic action in nature. In general, therefore, the magnetic field cannot be explained as a relativistic manifestation of the electric field !
  This relativistic combination of electric and magnetic forces will be analyzed in more detail in §1.6, part "Four-dimensional electrodynamics", where the electric and magnetic fields will be combined into a 4- tensor of electromagnetic field of 2nd order. We will see that by changing the inertial frame of reference, its electrical and magnetic components are mixed - similarly to the special theory of relativity, it mixes spatial and temporal coordinates in space-time. This is the theoretical basis of relativistic electromagnetism.

Electromagnetic waves
The general laws of wave origin and propagation in nature are discussed in §2.7, passage "Wave propagation - a general natural phenomenon". Here we show the origin and properties of waves in an electromagnetic field.
  The basic mechanism of the electromagnetic waves creation by the movement of electric charges was outlined at the beginning of this chapter :
  If electric charges move at a variable speed (unevenly, with acceleration or deceleration), they create a time-varying electromagnetic field around them, which leads to the creation of electromagnetic waves, that detach from their source and carry part of its energy with them into space. According to Faraday's law of electromagnetic induction, an electric field is created during movement or time changes in the magnetic field; and time changes in the electric field in turn cause a magnetic field. Thus, they mutually induce each other (however, we will see below that this claim is misleading).
  The uneven - oscillating - movement of electric charges (event. associated with an alternating voltage ~U) generates an oscillating electromagnetic field that propagates into the surrounding space at the speed of light c. The amplitude of oscillations in the electromagnetic wave thus created decreases quadratically with distance. During its propagation, a wave contains a mutually coupled and vectorally perpendicular electric E and magnetic B component, which are also perpendicular to the direction of propagation, of the speed of light c :
          
Note: It is often stated that according to Maxwell's equations (1.38) and (1.40) the electric E and magnetic B fields in a wave induce each other by the dynamics of their variability during oscillation. However, we will see below that this often stated statement is misleading. Sometimes we can even meet the statement that the mutual induction of the electric and magnetic components in an electromagnetic wave causes their propagation through space. This is not the case - the propagation of the wave is caused by the finite velocity c of the propagation of the electromagnetic interaction in space, by retarded potentials (1.47). The mutual induction of the electric and magnetic components can only possibly ensure the constant same structure of the wave during its propagation through space. In the figure in the middle, we have drawn the electric E and magnetic B components in the wave with the same amplitude for clarity. In reality, however, the magnetic component is incomparably smaller than the electric one - electromagnetic waves are in practice the electric waves - it is discussed below in the passage "Lesser-known properties of electromagnetic waves". Therefore, the "mutual induction" of E and B cannot actually occur effectivelly...
  When an electromagnetic wave encounters electric charges, it starts to oscillate them in the rhythm of its frequency (in the figure on the right), weaker or stronger depending on its amplitude (reduced in proportion to the distance from the place of its origin). This is the basis of the effect of electromagnetic waves on the material environment, as well as their laboratory detection and radio reception for the transmission of sound, image and digital-data information.
  The basic properties of electromagnetic waves follow from Maxwell's equations of the electromagnetic field. Therefore, if we write Maxwell's equations (1.38) and (1.40) for a vacuum spatial region, where j = 0 and r = 0, then by their partial derivation according to time and substituting from the remaining two Maxwell's equations we get d'Alembert's equations

analogous to equations (1.46) for potentials, but without the presence of electric charges. Since these equations have non-zero solutions, the electromagnetic field can also exist independently, without direct connection to electric charges and currents. If we look for particular solutions that depend on only one coordinate, eg on x , and at time t , equation (1.48) is simplified to

¶²E/¶x² - (1/c²) ¶²E/¶t² = 0 (and analogously for B)

and the solution will be every function of the shape

E = E(x, t - x / c) ,   B = B(x, t - x / c)   .    

The same value of the field E and B as in the point o coordinate x_o at the time t_o will be in all places whose coordinates and time satisfy the equation x - x_o = c.(t - t_o). It is thus a wave propagating in the direction of the X axis at the phase velocity c . The electromagnetic field in a wave varies as a function of distance and time.
  From the Maxwell's equations thus follows the existence of electromagnetic waves that propagate at a speed equal to the speed of light (from a general-physical point of view, the speed of light is discussed in §1.1, passage "Speed of light"). This finding led Maxwell to believe that light is probably an electromagnetic wave of very short wavelength. In this way, Maxwell managed to unify into a comprehensive theory not only electrical and magnetic phenomena, but also to include optical phenomena.
Note: The origin and properties of various types of electromagnetic radiation (radio waves, infrared radiation, visible light, UV and X-rays, g radiation) are discussed in more detail, for example, in §1.1 "Atoms and atomic nuclei", section "Electromagnetic fields and radiation" in treatise "Nuclear Physics and Physics of Ionizing Radiation".

 Subtle properties of electromagnetic waves
From the perfectly researched and verified laws of electrodynamics (summarized in Maxwell's equations), only three basic properties of electromagnetic waves clearly follow: 1.They are transverse waves (they oscillate perpendicular to the direction of their propagation); 2.Tthe electric E and magnetic B intensity vectors constantly oscillate perpendicularly to each other; 3.They propagate in a vacuum at the speed of light c. Other "more subtle" properties may be different, depending primarily on the specific way they are generated and on interactions with the material environment. We will mention here some more subtle properties of electromagnetic waves (on which sometimes different opinions appear).
-> The relationship between the magnitude of the electric E and magnetic B components in waves.
In various illustrations, pictures and diagrams, the electric and magnetic components are usually shown as approximately the same size. In reality, however, it is completely different. From relations (1.33)-(1.34) on the excitation of a magnetic field by charge and induction, it follows that a wave with an electric amplitude E is accompanied by a magnetic field with an amplitude B = E/c. The ratio of the amplitudes of the electric and magnetic fields in a wave is therefore
         ^E[V/m] /B_[Tesla] = c = 3×10⁸ [m/s]    .
Even very strong electromagnetic waves are therefore accompanied only by very weak magnetic fields (e.g. a relatively strong wave with an electric amplitude E ~1000 V/m will be accompanied only by a very weak magnetic field B ~3.3×10^-6 T, significantly weaker than the Earth's magnetic field). Electromagnetic waves are therefore practically in principle electric waves ! However, the numerical values of E and B depend on the system of units used.
Note: The above ratio E/B = c and the equation (1.56) of the distribution of electric and magnetic energy density below are valid only for plane monochromatic waves emitted by an oscillating electric dipole.
-> Relative phases of the electric and magnetic components.
Do these components oscillate in phase or with a phase shift ? In most diagrams in the literature, they are drawn in the same phase. However, when generating electromagnetic waves with a classical electric LC oscillator, oscillating at a frequency of f = 1/[2pÖ(LC)], the electric component in the inductive zone is expressed by the cosine time dependence E = E_o.cos(2p.t) and the magnetic component by the sinusoidal dependence B = B_max sin(2p.t). So there is a phase difference of 90°. There is no known way in which this phase difference would be cancelled later in the wave zone. However, in light of the above-mentioned fact about the very small representation of the magnetic component, this probably has no practical significance.
  For this reason, the often-stated statement that in an electromagnetic wave the electric and magnetic fields mutually induce each other loses its justification. Changes in the electric field induce a very weak magnetic field, which is certainly not capable of inducing an electric field of the appropriate intensity back.
-> Are the electric and magnetic components generated in an electromagnetic wave mutually ?
It is often stated that in an electromagnetic wave, a change in the electric field causes a magnetic field, the change of which then induces a changing electric field and so on - all the time, and this is the reason for the propagation of the wave. Spatially varying electric and magnetic fields can cause their mutual change in time, but Maxwell's equations and their solutions generally do not imply a causal link between the electric and magnetic fields. The electric and magnetic components are simultaneously and independently created by their common sources - the time-varying distribution of electric charges and currents. The acceleration of charged particles thus generates an electromagnetic field, which propagates into the surrounding space and then, at a greater distance, becomes independent of the source of the electric field that created it; it propagates as EM waves. Opinions on the causal relationship between the electric and magnetic components in electromagnetic waves vary, it is perhaps a question of interpretation..?..
-> Are all electromagnetic waves sinusoidal ?
Electromagnetic waves are almost always represented graphically as sinusoids. Such or a similar shape is expected given that EM waves most often arise from the vibrational-oscillating motion of electric charges, which has a harmonic "sin-cos" course. All oscillatory motions caused by a force F which is directly proportional to the deviation x from the equilibrium state F = -k.x, occur with the resulting deflection of the oscillatory shape x(t) = r.sin2p.f.t with a frequency f = Ö(k/m), where m is the mass of the particle.
  When we look at waves in the material environment in nature, they tend to be sinusoidal waves. If we stretch a guitar string at a certain point a little from its original straight line, it will take on a slightly triangular shape. When we let go of it, it returns and swing into the opposite shape and back again, but after several oscillations it will already oscillate sinusoidally at the fundamental frequency given by the length and tension of the string. Longitudinal elastic forces act in the string, which dilute the irregular oscillations and convert them into harmonics. Similar forces act in all elastic material environments. However, in a vacuum there are no such forces, so the oscillation of the fields here can be of very different shapes.
  However, in general, the time dependence of the oscillations E and B on time and position can be of different shapes, depending on the movement of the electric charges by which it is excited. If, for example, we were to introduce an alternating voltage of a rectangular or sawtooth waveform from a multivibrator into a transmitting antenna, a wave with this waveform E and B would be emitted, the character of which would be preserved during propagation through space. Only in the case of a harmonic sinusoidal waveform in the source is a sine wave emitted. However, each configuration of electromagnetic waves can be decomposed into a superposition of a smaller or larger number of harmonic sinusoidal waves of different frequencies and amplitudes using Fourier analysis. The sine wave is just a mathematical idealization, real waves always differ from this shape to a greater or lesser extent. Whether it is due to their fluctuations in the source - modulation in the transmitter, or interaction with the material environment, where their speed and behavior can change (even depending on the frequency). Sinusoids are therefore a mathematically advantageous set of functions that can be used to model a number of natural processes using simple harmonic oscillators. It is discussed above in the passage "Privileged sinusoids!".
-> The question of whether longitudinal electromagnetic waves could also occur in addition to transverse electromagnetic waves is occasionally discussed. When it comes to waves in a vacuum, waves other than transverse waves cannot arise there, because there is no medium whose particles would oscillate and, by interacting with other particles, transmit the oscillations further and further forward. In homogeneous and isotropic dielectric material environments, electromagnetic waves are also transverse. In inhomogeneous environments at the interface of materials with significantly different permittivities and permeabilities, as well as in birefringent dielectrics, electromagnetic waves can effectively (seemingly?) exhibit a weak longitudinal component. A specific situation is in a plasma environment, where longitudinal plasma waves commonly arise. However, these are not electromagnetic waves, but waves of density of charged particles (electrons and ions), which can secondarily create an electromagnetic field vibrating longitudinally and traveling at a speed significantly lower than c.
  More general question :
-> Can electric and magnetic fields exist separately ?
Under static conditions, not changing with respect to time, one can exist without the other. For example, a superconducting coil with a constant direct current has a magnetic field, but no electric field. Conversely, a charged capacitor disconnected from an electric source has an electric field without a magnetic field. We do not consider here microscopic quantum oscillations and magnetic moments of electrons and protons.
  However, under dynamic conditions of time variability, the electric and magnetic fields cannot exist separately. According to Maxwell's equations (1.38) and (1.41), their variability mutually generates each other. If one field is time-varying, the other must be non-zero and also time-varying.

Targeted transmission and reception of electromagnetic radio waves
The uneven movement of electric charges represents only the basic conceptual mechanism of wave generation. For the targeted creation of sufficiently strong electromagnetic waves for applications in communications technology, industry and scientific research, it is necessary to effectively generate these waves using electronic methods. It is necessary to create intensely vibrating systems of a large number of charges in electrical circuits - alternating electrical voltage and current. We will introduce the basic methodology of transmitting and receiving electromagnetic waves for classical radio technology in the area of lower and medium frequencies of approximately 300 kHz - 3 MHz (wavelengths of 1000 m to 100 m - medium waves), for the 3 - 30 MHz band (wavelengths of 100 to 10 meters) and very short radio waves of frequencies of 30 - 300 MHz. We will then mention higher frequencies in the following passage "Electromagnetic waves of very high frequencies".
 Electric oscillator
The basic starting electronic circuit is an electric oscillator. In the simplest embodiment, it is formed by a capacitor of suitable capacitance C and a coil of suitable inductance L connected in parallel, in connection with an active element (usually a transistor) creating feedback and supplying electrical energy U.
  If a capacitor C charged to a voltage U is connected in parallel to a coil L, the capacitor will start to discharge through the coil, a current will start to flow through the coil and a magnetic field will be created around it. After the capacitor is discharged, the current will stop, but in the coil, the accumulated magnetic field will start to induce a voltage of the opposite direction, which will start to charge the capacitor back. And after it is charged, the discharge process will repeat. The capacitor and the coil will periodically exchange energy with each other with a frequency f = 1/[2p.Ö(L.C)]. Since there are always ohmic and radiation losses in the circuit, the amplitude of the oscillations would decrease exponentially, the oscillations would be damped. If these losses are continuously replaced electronically by an amplifier in feedback, the amplitude will not decrease, and a stable source of alternating voltage will be created.
  Therefore, in this resonant circuit LC, alternating charging and discharging of the capacitor through the inductance of the coil will create sinusoidal electrical oscillations with a frequency f = ¹/2p.Ö(L.C) and a period T = 2p.Ö(L.C), which will be maintained by feedback and then amplified. By changing the values of L or C, the frequency is regulated - tuning is performed; usually tuning is performed by changing the capacitance C using a variable tuning capacitor (rotary or electronic). Instead of a tuned circuit, a crystal piezoelectric resonator is sometimes used in oscillators (it allows high accuracy and frequency stability).
  The simple Thomson LC oscillator is no longer used in transmitters, but more complex electronic oscillators have been developed. .... ....... .........
  A simple LC resonant circuit is used in receivers, most often in superhets. Tuning is done by changing the capacitance C using a variable tuning capacitor. Previously, a mechanical rotary capacitor was used (in the picture on the top left), in the medium and short wave bands, a rotary tuning capacitor in the range of 0÷500 pF and an inductor of about 0.1 mH (about 50 turns in air without a core, for higher frequencies with a correspondingly lower number of turns). Instead of a rotary capacitor, an electronic varicap is now used.... ...
  For the needs of transmitting information - sound, music, images, digital data - the basic high-frequency voltage is modulated in amplitude or frequency. In the rhythm of the required transmitted signal, its amplitude or frequency increases or decreases slightly. This signal is then amplified in the power output circuit of the final amplifier, which is equipped with power tubes or power transistors. At higher powers, effective cooling is also installed.
 Antenna
When we have such an alternating voltage source, it is necessary to bring it to an external conductor, which will create an alternating electric or magnetic field around it, which will then spread into space. Such a conductor is called a (transmitting) antenna. In the simplest case, it can be an ordinary piece of wire or a metal rod; the second pole of the electrical circuit is the earth - grounding (from an electronic point of view, the antenna is modeled by a so-called electric dipole).
  Antennas are radio-technical systems of conductors that are capable of radiating electromagnetic waves into free space (transmitting antennas), or, conversely, receiving these waves from free space and converting them into electrical signals (receiving antennas). When a transmitting antenna is powered by an alternating voltage of frequency f, it converts part of the supplied electrical energy into the emission of electromagnetic waves of length l = c/f. Antennas are characterized by several basic parameters :
-> Antenna transmission efficiency, or sensitivity of the receiving antenna. It is optimized by suitable antenna dimensions and geometric arrangement of individual conductors; its length should be at least l/2 .
-> Directional selectivity - antennas transmit or receive better in certain directions than in others. This property is quantified by the directional radiation characteristic of the antenna.
-> Frequency selectivity, when antennas have different efficiency for different frequencies f or wavelengths l. For shorter wavelengths, polarization selectivity is also be applied - the geometric design and orientation of the antenna in space affect how effectively it receives waves of different polarizations.
-> Antenna impedance. From an electronic point of view, an antenna is part of an electrical circuit with an alternating signal - the transmitting antenna is the last (output) element of the transmitter, the receiving antenna is the input element. Therefore, it depends on how it behaves electrically in this circuit, i.e. what its resistance is, in this case, the impedance Z = U/I. It is the ratio of voltage U and current I at the antenna connector. For example, a half-wave dipole antenna has an impedance of 75 O, a Yagi antenna has Z=300 O. . .......
.....picture of radiation characteristics of different types of antennas ....???
  Each elementary section of the antenna radiates waves in all directions, while a different current flows at each point of the antenna. So waves with a different intensity are radiated from each section. The radiation of the entire antenna is then given by the sum of the individual elementary sections. In general, all radiation characteristics strongly depend on the geometry of the antenna and on the ratio of its size and wavelength.
  The electromagnetic field changes its character at different distances from the transmitting antenna. For an antenna of size D, which transmits at wavelength l, we distinguish three significant zones according to the distance (radius) R from the antenna :
-» The closest area, sometimes called reactive, is in close proximity to the antenna 0<R<l, to a distance smaller than the wavelength. In this immediate vicinity of the transmitting antenna, electromagnetic fields of complex waveforms arise, which do not yet have the transverse character of electromagnetic waves. The electric field has a longitudinal propagation component in addition to the transverse component. There are constant spontaneous "reactions" - energy exchanges between the variable fields and the source, the neighboring antenna. There is an oscillation "back and forth", unrelated to radiation.
-» In the inductive zone, also called the near zone, at a distance in the range l < R < 2.D²/l, the electromagnetic field already has a transverse wave character, but inductive feedback with the antenna still persists here. The radiated power here depends to some extent on the presence of a "receiver" of the emitted waves (placing the receiver in the inductive zone will somewhat increase the electrical power input of the transmitting antenna).
-» In the wave zone, also called the distant zone, at a distance of approximately R > 2.D²/l , in practice typically from several tens of wavelengths, a perpendicular course of the oscillating electric and magnetic field is established in a wave with a locally planar wavefront (in large areas of space the wave has a spherical wavefront) and the wave energy is irreversibly radiated into free space, independently of the presence of any "receiver". There is no longer any feedback with the transmitting antenna.
  To achieve a long range, the antennas of powerful transmitters are usually of robust construction placed high on masts. The high-frequency signal is fed to the antennas using coaxial cables. The transmitting power of large professional transmitters is sometimes several tens of kW to megawatts. In the medium and short wave range, wave reflection in the ionosphere can also be used for long-range transmission/reception.
       
  The receiving antenna can be of a similar design to the transmitting one, but in principle much smaller. Electromagnetic waves induce in it a very weak signal of the same frequency and waveform as the one sent. The received frequency is again tuned in a resonant LC circuit. The selected signal must then be amplified many times over and then it can be demodulated - extract the signal carried by changes in amplitude or frequency - and used for communication or research purposes.
 Superheterodyne radioreceiver
Simple receivers with direct amplification of the RF signal from the receiving antenna (as in the picture on the right) had low sensitivity and insufficient frequency selectivity. Therefore, significantly more complex electronic processing of the received signal was developed. The so-called superheterodyne receiver (the name is a combination of superposition and Greek hetero=different, dyn=power, strength.....), abbreviated as superhet, superimposes the received signal frequency with the tunable frequency of the local oscillator into a difference intermediate frequency, which is then amplified and demodulated. The local LC oscillator is tuned using a double rotary capacitor or varicap, synchronously with the receiving LC resonant circuit. By mixing the received signal with the frequency of the local oscillator, a sum or difference frequency is created. Over the decades, a standard has emerged that for the frequency range of the received signal of approximately 100kHz-30MHz (medium, long and short waves) an intermediate frequency of mostly 455kHz is used, for VHF frequencies of approximately 30-300 MHz an intermediate frequency of 10.7MHz is used. The intermediate frequency signal is then amplified, frequency-selected using narrow intermediate frequency filters and then demodulated into the resulting audio or video signal. The superhet thus achieves high sensitivity, signal-to-noise ratio and high frequency selectivity.
 Author's apology :
I wrote the passage about the transmission and reception of radio waves here somewhat more technically focused than is necessary for the physical explanation of electrodynamics, gravity, space astrophysics. For me, it is a kind of reminiscence of my school and grammar school years, when electricity and radio engineering fascinated me. Already in the 8th and 9th grades of elementary school, I constructed two radiotransmiters myself: for the range of about 0.5 - 30 MHz and another around 70 - 100 MHz, equipped with EF22 and PL36 tubes. And as a radio amateur, I tried to receive distant stations - DX (Distance reception at X-distance, as large as possible; I even succeeded from China). I apologize to any readers (otherwise oriented) for this subjective atypia and thank you for your understanding...
Electromagnetic waves of very high frequencies
To generate electromagnetic radiation of very high frequencies of the order of GHz (e.g. 3-30 GHz) and therefore short wavelengths of the order of centimeters (e.g. 10cm - 1cm), classic LC oscillators can no longer be used, but instruments of high-frequency electronics. Oscillators for these high frequencies are basically of two types :
-> Special vacuum tubes, magnetrons and klystrons with resonant cavities (described in more detail in §1.5, passage "Electrical power supply of accelerators"). They are mainly used for large high-frequency powers of the order of kilowatts to MW.
-> Special semiconductor components with negative differential resistance, such as Gunn diodes, tunnel diodes, IMPATT diodes (impact ionization avalanche diode). If such an element is integrated into the circuit of a microwave resonant cavity and a DC voltage U is applied (approx. 9V for a Gunn diode) to achieve the negative resistance region, spontaneous oscillations begin to occur (in the figure on the left). The frequency then depends on the size of the resonant cavity.
  For conducting ultra-HF oscillating voltage from the oscillator to the antenna, coaxial cables (where there are high losses) are no longer optimal, but waveguides - metal tubes of rectangular or circular cross-section are often used. When an electromagnetic wave strikes the conductive walls of the tube obliquely, it is completely reflected from them. It propagates in the longitudinal direction through the waveguide by successive reflections. The superposition of the incident and reflected waves creates a complex waves in the waveguide that has a component of a progressive wave in the longitudinal direction and a standing wave in the transverse direction.
  In the simplest case, transmitting antennas for centimeter radio waves can again be ordinary wires or rod antennas with omnidirectional effect. However, due to the short wavelengths, there is the possibility of narrowly directional antennas in the shape of funnels or paraboloids :
        
  These very short waves propagate in a straight line - only in direct visual visibility without obstacles, there is no bending or reflection from the ionosphere. Receiving antennas can again be wire (rod) without directional effect, or directional parabolic.
  For our astrophysical sciences, radio telescopes (pictured on the right) and detection of microwave cosmic rays are important (§5.4, passage "Microwave relic radiation - messenger of news about the early universe").
  For even higher frequencies, in the infrared, visible and UV radiation range, no electronic oscillators and antennas work anymore, but this radiation is created by vibrations of excited atoms and molecules in substances. Special techniques are used here combining electronic methods of atomic physics (including laser), with photomultipliers and semiconductor photodiodes for detection and spectrometry of this radiation.
And the highest frequencies and energies of X and gamma radiation are already the domain of nuclear and radiation physics (detailed discussion "Nuclear physics and physics of ionizing radiation").
  Electromagnetic waves are very important in astrophysics, as they arise from a variety of processes in the universe (in principle, almost all of them, even in the vicinity of black holes). And they allow us to investigate and indirectly "peek" into these often extremely distant processes in the depths of the universe - it is discussed in §1.1, passage "Electromagnetic radiation - the basic source of information about the universe".

Plane electromagnetic waves
In a plane wave propagating in the direction of the X axis, all quantities are functions only t-x/c. If E = E (t-x/c), then from Maxwell's equations (1.38) and (1.40) for r = 0, j = 0, it follows ¶B/¶t = -rot E = (n°/c) ´ (dE/d(t-x/c)) = n°´ ¶E/¶t, so the relationship between electric and magnetic field in an electromagnetic wave is

where n° is the unit vector in the direction of wave propagation ("x" indicates the vector product). That is, the vectors of electric and magnetic fields E and B are always perpendicular to each other and also to vector n° direction of propagation - electromagnetic waves are transverse. Since B = rot A, only the vector potential A is sufficient for the description of the plane wave, by which the fields E and B are determined by the relations

(the dot above A means the time derivative: A^. = ¶A/¶t).
  The simplest case of an electromagnetic wave is a monochromatic wave, in which the field is a simple harmonic function of time at each given point: A(t)_{r = const.} = A_o(r) .cos(wt + a), a = a(r), where w = 2p.f = 2p/T is the circular frequency of the wave, a is a constant phase shift. Size l = 2p c/w then represents the wavelength, ie the distance that the wave travels in one period T (the distance of the two nearest places with the same phase). In a planar monochromatic wave, the field will be a harmonic function of the argument t-x/c

A = A_o cos [w.(t - x/c) + a] ,     

where A_o nor a does not depend on t nor x. By introducing a wave vector

a plane wave can be expressed in form

valid for any direction of wave propagation (analogously for B and E). This expression for a monochromatic plane wave is often written in a imaginary complex form

where Â_o = A .e ^{i a} is a constant complex vector; the fields E and B can be expressed similarly.

The speed of propagation of an electromagnetic signal
The commotion - signal - in the electromagnetic field therefore propagates in space with a finite speed c, resulting from the retarded potentials (1.47a,b), which are solutions to Maxwell's equations. We can measure the propagation speed of electromagnetic signals in basically three environments :
-> In a vacuum, where it is the speed of light in a vacuum c = 299,792.458 km/s. - the maximum possible speed of propagation of interactions in nature (§1.1, passage "Speed of light").
-> In a dielectric material environment, where this speed c´= 1/Öem is somewhat smaller than c=1/Öe_om_o in a vacuum (§1.2, passage "Electromagnetic and optical properties of substances"). In air (at normal pressure) it is 0.9997 c = 299,702 km/s, in water about 0.75 c = 225,000 km/s, in diamond only 0.413 c = 123,800 km/s.
-> In metal conductors - electric wires - we measure the speed of movement of the electrical signal along the conductor (we are no longer talking about electromagnetic waves that do not penetrate inside the conductor), which is determined, in addition to the speed of light in the given environment around the conductor, also by the specific inductance and capacity of the conductor . For a straight copper wire (cable thickness ~ 1 mm) approx. 0.65 c ~ 195,000 km/s was measured. It is worth noting that the speed of propagation of an electromagnetic signal in conductors somewhat depends on whether it is a "bare" wire without insulation, or an insulated cable, or coaxial.
  In electronic circuits and networks, the time delay of the signal is important, given by dividing the length of the cable by the speed of the signal. It limits the minimum width or maximum frequency of pulses that networks and circuits are able to transmit and process. In high-frequency electronics, phase shifts between voltage and current and unwanted reflections and signal interference occur in conductors. These effects of reflections and signal interference can also significantly affect electronic measurements of signal speed in conductors.
  As a result of these effects, there are above all artificially higher - superluminal - measured signal propagation speeds in electrical cables, from which sometimes rather bombastic claims are derived such as "I measured the superluminal speed of signal propagation in an electrical cable - and thus disproved the theory of relativity! "...

Internal angular momentum - spin - of electromagnetic waves
When turn of the coordinate system by an angle J around the direction of propagation n° of plane electromagnetic waves, the field in the wave will be transformed according to the law  ® Â' = e^iJ. Â; the electromagnetic wave is invariant due to the rotation of an angle of 360° around the direction of propagation. The symmetry properties of plane waves with respect to rotation around the direction of propagation are important in quantum physics, where they determine the spin of the respective particles created by quantization of a given field. At the classical level, spin is defined as

s = 360° / (angle of symmetry of a plane wave with respect to rotation about the direction of propagation) ;          

the spin of the electromagnetic waves and their quanta - photons is therefore equal to s = 1 .

Orbital angular momentum of an electromagnetic wave. Helically polarized radiation.
Momentum p is a quantity quantifying the "intensity of motion" of an object of mass m moving at speed v: p = m.v. Angular momentum of momentum L is then a quantity describing how this movement is oriented relative to space - to reference points, usually to the origin of the coordinates: L = r x p, where r is the vector of the current distance of the particle from the origin of the coordinates.
When a beam of electromagnetic radiation arrives, in addition to energy, it usually also has a certain angular momentum relative to the origin of the coordinate system. During interaction, it is transferred together with energy (all or part of it) to a given particle or system....
  When electromagnetic waves are emitted by a rotating source, this rotation leaves specific traces on the structure of the emitted waves. During the formation of a wave, the maximum and minimum E and B shifts periodically in time relative to each other during the rotation of the source, so the phases of the wave - the alternation of maximum and minimum - will shift as the wave propagates at speed c. The wavefront takes on the helix shape of a spiral. A kind of "rotating, twisting or spiral radiation" with specific polarization properties is created. If it hits charged particles, it causes, among other things, their rotation around the direction of propagation of the beam, which can be considered as a manifestation of the angular momentum of this radiation. So it behaves like a wave with orbital angular momentum.
  In §4.4, the passage "Light propagation in the field of a rotating black hole" it is discussed that a similar situation can occur in rotating black holes, where the radiation from the accretion disk can be spirally polarized in this way due to the rotational deformation of spacetime.
Note: In laboratory optical experiments, this light with helically rotating polarization is generated by special rotating sources, cylindrical lenses, lasers, holograms, and measured using spatial light modulators. For radio waves, helical polarization can be achieved by a special spatial configuration of antennas powered by phase-shifted alternating signals, for sound waves analogously by speaker systems. Gravitational waves arising in rotating systems of black holes or neutron stars should also have a component of helical rotating polarization (however, their measurement is complicated, so far it has not been possible).

The general laws of wave origin and propagation in nature are discussed in §2.7, passage "Wave propagation - a general natural phenomenon".

Energy of electromagnetic field
In electrostatics, it can be shown by simple considerations (about the work needed to place the charges in a given configuration) that the electrostatic energy of a system of N charged bodies

e_e = (1/2) _a=1S^N q_a.j_a = (1/2) nr.j dV = (1/8p) n E² dV       

can be expressed by the integral of the intensity of their common electric field, so that the electric field can be assigned the energy distributed with the density W_e = (1/8p) E² in space. Similar considerations of labor required to produce the electric currents in the system of electrical circuits (against induced electromotive forces generated increase of the magnetic field), show that the energy of system this conductors

e_m = (1/2) _a=1S^N I_a.F_a = (1/2) nA.j dV = (1/8p) n B² dV       

is given by the volume integral of the induction vector B of the excited magnetic field and can be considered as the energy of this magnetic field distributed in space with density W_m = (1/8p) B². The energy density in the electromagnetic field is then equal to the sum of the densities corresponding to the electric and magnetic components :

It is clear that such an assignment of energy to the field is purely formal under Coulomb, Ampere, and Faraday's law, because it is just another description of the interaction energy in the idea of the instantaneous force action of charges and currents at a distance. However, the physical justification is given by the fact that the commotion in the electromagnetic field propagates at a finite speed. This final rate of propagation of changes in the field leads to the conclusion (repeat, see the argument in the introduction §2.8) that the electromagnetic field itself must actually contain energy (and momentum) that can flow from one place to another and perform work on electric charges and currents - changing to other forms of energy. The electromagnetic field is therefore not just a space in which electric and magnetic forces act, but is a separate physical reality - a specific form of matter.

By scalar multiplication of Maxwell's equation (1.38) by the field E and equation (1.40) by the field B and their addition we get the equation after adjustment

Integration over some chosen spatial area V after the application of the Gauss theorem then gives

(1.54)

The left side represents the change in the energy of the electromagnetic field e_elmag contained within the region V per unit time. The first integral on the right indicates the work that electric forces do with charges per unit time, or the change in kinetic energy e_kin of charges per unit time (magnetic forces with charges do no work and therefore do not change their kinetic energy). Equation (1.54) thus expresses the law of conservation of energy in the electromagnetic field: the electromagnetic energy contained in the spatial region V decreases on the one hand by mechanical work performed by electric forces with charges inside the region V , on the other hand by energy transmitted (radiated) by the field from the area V through the bounding surface S = ¶V to the outer space. Equation (1.54) can also be written in the form

(1.54 ')

whereby a decrease in total energy of the electromagnetic field and the charged particles in the volume V per unit time is equal to the flux vector (c/4p). (E x B) by the surface S surrounding region V. Therefore the vector

called the Poynting vector, represents the energy passing through a unit of area per unit of time, or it is a vector of the flux density of electromagnetic energy in space. When integrating in (1.54) over the whole space, when the bounding surface S is infinitely spaced and the field on it is equal to zero, it expresses equation (1.54), resp. (1.54'), simply the law of conservation of the sum of the total energy of the electromagnetic field and the kinetic energy of all charges.
  Similarly, it can be shown that the electromagnetic field has momentum p given by the integral

so the momentum of the volume unit of the electromagnetic field is equal to P/c² .
 Energy transmitted by electromagnetic waves
Electromagnetic waves carry a certain part of the energy from the source system into the surrounding space. At a sufficiently large distance - in the wave zone - they already exist independently, without being connected to the original source, which is then no longer affected by whether the wave has transmitted its energy somewhere. The electromagnetic wave itself contains energy regardless of whether it is absorbed or not. When a wave encounters a system of electric charges, its oscillating electric and magnetic fields can exert forces and move these charges - perform work, transmit energy. A greater or lesser part of the wave's energy is thereby absorbed, the intensity decreases and possibly the remaining part travels further.
  The greater the intensity of the electric E and magnetic B fields - the amplitude of the wave - the greater the energy the wave carries and the greater the effect it can have on the electric charges. The energy flow density in a plane electromagnetic wave is, with respect to (1.49), equal

which with respect to (1.52) is related to the energy density W_elmag by the relation P = c. W_elmag . n°, from which it can also be seen that both the field and energy propagates in the wave at the speed of light.

The density of electric energy in an electric field is W_e = (1/2)e_o E². The density of magnetic energy in a magnetic field is W_m = B²/(2m_o). The energy at any place and time in an electromagnetic wave is given by the sum of the energies of the electric and magnetic fields, i.e. for densities :

W_elmag  =   W_e + W_m  =   ¹/₂ e_o E² + ¹/_2mo.B²   .

In an electromagnetic wave, E and B are related by the relation E = c.B, the speed of light in a vacuum is 1/Ö(e₀.m₀). So in an electromagnetic wave, the electric and magnetic energy densities are related :

i.e. the magnetic energy density W_m and the electric energy density W_e are the same here, despite the fact that a changing electric field generally produces only a weak magnetic field. The total energy of the wave is then W = W_e + W_m = e_o E² . The intensity of the electromagnetic wave, which is the energy passing through a unit area (perpendicular to the direction of propagation) per unit time, is then I_W = c.e_o.E² .

Electromagnetic energy radiation
Let us have a system of moving electric charges concentrated in some limited spatial area (Fig.1.4). If we place the origin of coordinates somewhere into the system of charges, then the study of the field at large distances R >> L, where L is the characteristic dimension of the system will place all of the source system at about the same distance R as the origin of the coordinates. Distances |R - r| of the individual points r' of the source from the investigated distant point R is approximately equal to |R - r'| @ R - R°. r', where R° is a unit vector pointing from the origin O to the investigated point, so that the retarded potential can be written in the form

j(R,t) = (1/R). n r(r', t - R/c + R°.r'/c) dV' , A(R,t) = (1/R). n j(r', t - R/c + R°.r'/c) dV' .

The retardation time thus consists of two different parts. The first part R/c determines the external retardation, ie the time required for the changes in the electromagnetic field to exceed the distance from the origin of the coordinates, or from the source system, to the distant observation point. The second part equal to - R°. r'/c characterizes the internal retardation, ie the time of propagation of the disturbance in the field within the source system.
If the charge distribution in the system changes slowly enough, internal retardation can be neglected. For this, it is sufficient that the characteristic time T, during which the charge distribution changes appreciably, satisfies the condition T >> L/c. Since c.T is the wavelength l of electromagnetic waves emitted by the system, the condition of negligibility of internal retardation can also be written in the form L << l, ie the dimensions of the system must be small in comparison with the length of the emitted waves. The characteristic time T changes in the distribution of charges is related to the average velocity v of charges rounds following equation: T » L/v, so that neglecting retardation it is necessary to apply v«c, i.e. the speed of motion of charges must be small compared to the speed of light. If internal retardation is neglected, the potentials are equal at great distances from the source system

j(R,t) = (1/R). n r(r', t - R/c) dV'   ,   A(R,t) = (1/R). n j(r', t - R/c) dV'   .    

At these distances large in comparison with both the dimensions of the source system and the length of the radiated waves - in the wave zone - it is possible to consider the variable field component as a plane wave within small areas of space. It is therefore sufficient to determine the vector potential A = (1/cR). n r.v dV' = (1/cR)_a=1S^Nq_av_a = (1/cR) (d/dt)_a=1S^Nq_ar'_a , i.e.

where d º S q_a r_a is the electric dipole moment of the system as it was at time t-R/c. The electric and magnetic fields are then, according to (1.49), equal

where the dipole moment d is again taken at the moment t-R/c (dots above d mean the derivative according to time).

The flow of electromagnetic energy in the wave zone, ie the intensity of electromagnetic radiation, is expressed by the Poynting vector according to (1.57)

where J is the angle between the directions of the vectors d^.. and R (if we use polar coordinates - Fig.1.4b). The angular distribution of the intensity of the electric dipole radiation is given by the coefficient sin²J , the corresponding directional diagram is in Fig.1.4c. The total energy radiated by the system per unit time (ie radiated power) I = dE/dt is then given by the flow of energy over the entire spherical surface R = const. :

(1.61)

If the source system consists of only one accelerating charge q, is d^.. = q. r^.. = q. a, and the radiated power is equal to

This radiation law was derived in 1899 by the Irish physicist J.Larmor. In addition, in the system of SI units, there is a coefficient k = 1/ (4p e_o) present in Coulomb's law.

Fig.1.4. Electromagnetic field of an island system of moving electric charges.
a ) The field excited by a system of moving electric charges is given not by an instantaneous but by a retarded distribution and movement of the charges.
b ) At a great distance from the source system (in the wave zone), the variable component of the field is given by the second time derivative of the dipole moment of the system d^.. and has the character of electromagnetic waves carrying the kinetic energy of the source into space.
c ) Directional diagram of electric dipole radiation.

Relationships (1.58) to (1.61) for the field and radiation of the island system of electric charges in the wave zone were obtained in the first order approximation in the ratio L/l (higher order members were neglected), which led to the application of only the dipole moment of the system. In the general case, however, it is necessary to take into account other members in the development of the potential according to powers L/l, which leads to the fact that the total intensity of electromagnetic radiation of a system of moving charges is given by time derivatives of individual multipole moments of charge distribution. In addition to the dipole moment is usually the most radiation involved the quadrupole moment K_ab = n r.(3x_ax_b - d_ab.r²) dV and possibly magnetic dipole moment m = (1/2c) n r.(rxv)dV, which contribute to radiation according to a known relationship (see eg [166])

(1.62)

If the properties of the source system are such that d^.. = 0 (this is the case, for example, in a system composed of bodies with the same specific charge q/m), dipole radiation does not occur. In such cases, only radiation caused by other members in the development of the potential according to powers of L/l , ie radiation of higher multipoles, is applied.

Electrodynamics thus comes to the general conclusion that with each accelerated (uneven) movement of electric charges, electromagnetic waves are emitted, which carry part of their kinetic energy into space *). In §2.7 we will see that the general theory of relativity arrives at essentially the same conclusion - the emission of gravitational waves during the accelerated motion of gravitational bodies, although the properties of gravitational waves differ from the properties of electromagnetic waves in some respects (above all, they are many times weaker).
*) This phenomenon plays an important role in atomic physics for the structure of the atomic shell and the formation of radiation during its deexcitation (see §1.1 "Atoms and atomic nuclei" of the book "Nuclear Physics and Physics of Ionizing Radiation"). Furthermore, in nuclear physics and physics of ionizing radiation. Particularly fast-flying electrons are sharply braked when interacting with the material environment, so that according to the relation (1.61´) they emit relatively intense electromagnetic radiation - so-called braking radiation. Braking radiation finds significant use in the excitation of X-rays by the impact of electrically accelerated electrons on the anode in X-rays tubes - see §3.2 "X-ray diagnostics", or in the excitation of hard g- radiation by the impact of high-energy electrons from betatron or linear accelerator on a suitable target (see §1.5 "Elementary particles", the section "Charged Particle Accelerators" of the same publication) .

We investigated the variable electromagnetic fields excited by a system of moving charges in the wave zone, ie at sufficiently large distances from the source system, and we calculated the radiated energy using the Poynting vector. Analysis of the electromagnetic field at short distances then shows that a small variable component of the electric field with a phase different from the main variable component is generated inside and near the source system. In the third order approximation, this term is equal

E_re  = (2 / 3c ³ ) d ^...  .    

In the source system, therefore, each charge q will be subjected to a certain additional force "reaction" f_re = q. E_re acting per unit time of work f_re.v so that the total work performed by this field with all charges of the system system comes out A _re = (2/3c³) d^... S q_av_a = (2/3c³) d^... .d., which when averaged over time (over several periods T) gives

A _re  = - (2 / 3 c³ ) d^{.. 2}  .    

It can be seen that this additional field component causes a corresponding braking of the charge movements in the source by the back reaction of the emitted waves, in full energy agreement with the formula (1.61) obtained by the analysis of the field in the far wave zone. Such an analysis is of great importance for gravitational waves, where the calculation of energy in the wave zone is not nearly as clear and unambiguous as in electrodynamics - we will see this in §2.8 "Specific properties of gravitational energy".
   Equation of motion m. v = q. E + (q/c). ( v x B ) charged particles in the electromagnetic field under the influence of the Lorentz force (1.30) is therefore necessary to supplement the braking action of electromagnetic radiation :

this equation is applicable when the velocity is small compared to the speed of light and the braking force is significantly smaller than the Lorentz force acting on the charge of the external field E and B.
   Further details on the properties of the electromagnetic field and their applications can be found in the relevant literature; from review momographies we can mention eg [235], [264], [206].

E t h e r
Electromagnetic fields in the past have been considered as a manifestation of certain types of ether motion *). Some (electrically charged) bodies set this ether in motion, which propagates in it at a finite speed and is passed on to other bodies. However, such an ether would have to have very unusual physical properties. In order for electromagnetic waves to propagate in it, which are transverse, it would have to have some properties of a solid body. And the mechanical model of the ether is no longer compatible with the experimentally determined constant of the speed of light in all inertial systems. Attempts to reconcile this fact with the ether model did not lead to success (for example, the assumption of "entrainment of the ether" by the movement of the Earth failed to confront the observed aberration of light from the constellations). Therefore, the idea of ether was abandoned and it was realized that the carrier of the electromagnetic field is the space itself . A. Einstein then completed this concept in a special theory of relativity by concluding that the stability of the speed of light is a reflection of the connection between space and time. Electromagnetism thus played an important heuristic role in revealing the deeper and more general laws of nature - the laws of relativistic physics .
*) Ether :  
According to ancient Greek philosophers and naturalists (especially Aristotles), the "heavenly" space - the universe - was filled with a hypothetical ubiquitous invisible subtle substance called ether (Greek aithér, Lat. ether = bright light, sky, upper layers of air). They imagined the ether as a flexible, transparent liquid, perfectly permeable without friction, unweightable, indestructible. Together with the other elements "earth, water, air, fire" as the "fifth element", which does not act on the earth's surface, but in the celestial spheres. All celestial bodies, stars, planets, the Sun are created from the ether. Ether transmits light and heat from the Sun and light from stars and planets.
Note: In chemistry, aether is called a volatile organic solvent.
   19th century physics she took it for granted that every wave could propagate only in the flexible material (matter) medium whose oscillating motion created it. It is difficult to imagine sea waves without water or sound without air (or other flexible acoustic environment of gaseous, liquid or solid phase - see the well-known elementary experiment with an alarm clock or a bell under the recipient of the vacuum pump). When it was discovered that light and other electromagnetic waves propagate not only in air and other optical media environments, but also in a vacuum, a problem arose in the environment or medium in which electromagnetic waves propagate. So once again revived the idea of ether - universal all- pervading "substances", filling all space and penetrating all matter (just as water penetrates the meshes of a fishing net towed behind a boat). This ether creates an environment for the propagation of light, heat and other electromagnetic waves; it is also a carrier of gravity. Since the ether did not manifest itself in any other physical and chemical phenomena, it was judged to be translucent, unweightable, perfectly permeable without friction, and had no chemical properties. A substance with such conflicting properties was practically undetectable experimentally.
   One could only study how the penetration of ether affects the speed of light under different configurations of the state of motion of the light source and the observer. Maxwell himself has already designed an experiment using the motion of the Earth: light moving with the ether in the same direction as the Earth orbits the Sun must have a slightly different speed than light that propagates perpendicular to this motion or in the opposite direction. be about 10^-7. Maxwell did not see the result of this experiment; up to 8 years after his death, in 1887, A.Michelson and E.Morley made this measurement by interfering with the rays of monochromatic light reflected by two mirrors in the horizontal and vertical directions, while the whole interference device on the floating plate could be rotated. The result was that no difference was measured in the speed of light in both directions, seemed unexpected and paradoxical at the time. However, the negative result of this experiment was confirmed by other measurements. No ad hoc hypotheses, such as ether entrainment (the ether is drawn along with the earth's surface, so its position relative to the interferometer) has not been confirmed. In contrast, the negative result of Michesson's and Morley's experiment was explained by the Lorentz contraction hypothesis, according to which the dimensions of all bodies in the direction of their velocity v are shortened in the ratio 1/ Ö (1-v²/c²). Einstein then gave a definitive and universal explanation in his special theory of relativity, according to which the speed of light (in vacuum) is the same in all moving conditions and in all directions. The idea of the ether was thus definitively abandoned, replaced by the properties of empty space itself, connected with time into a single space-time continuum. Nevertheless, in the field of radio applications of electromagnetic waves, the terms "transmit to ether" or "receive from ether" are often used.
   However, after the creation of general relativity, Einstein recognized that the role of the earlier static ether is equivalently played by the space-time continuum, which is dynamic and is described by the metric tensor field g_ik .

Nonlinear electrodynamics ?
At all the intensities we observe in nature and in the laboratory, the electric and magnetic fields in the vacuum appear to us to be linear - the principle of superposition applies exactly to the values of intensities E and B, as well as to potentials.
At the end of the following §1.6 "Four-Dimensional Spacetime and Special Theory of Relativity", passage "Nonlinear Electrodynamics", the possibilities of how an extremely strong electromagnetic field could behave in a non-linear manner even in a vacuum will be discussed.

1.4. Analogy between gravity and electrostatics		1.6. Four-dimensional spacetime and special theory of relativity

Vojtech Ullmann