# Electrodynamics, Maxwell's equations

 AstroNuclPhysics ® Nuclear Physics - Astrophysics - Cosmology - Philosophy Gravity, black holes and physics

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. Electromagnetic field. Maxwell's equations.

1.6.
Four-dimensional spacetime and special theory of relativity

1.5. Electromagnetic field. Maxwell's equations.

The most important force that determines all the 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 neutrons, 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 both the special and general theory of relativity.
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".

In an electromagnetic field acts on a test particle with a charge q moving at a velocity v the total force (Lorentz force)

 F =   q . E   +   q . (1/c) [ v x B ] , electric force       magnetic force (1.30)

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. 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. The motion of electric charges is described by 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

 div j   +   ¶r / ¶ t = 0 , (1.31b)

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

Act excitation electric field by electric charges, i.e. fundamental Coulomb's law (1.22a) in §1.4 above, can be expressed in the form of Gaussian theorems of electrostatics (Fig.1.3a) (1.32a)

from which the differential equation follows

 div E   = 4p r   . (1.32b) 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 e 2 c 3 / eh = 1.32 . 1016 V/cm, where me 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 c 2 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 theoretical model of classical relativistic nonlinear electrodynamics will be discussed.

The magnetic field is excited by moving electric charges, ie electric current, according to the Biot-Savart-Laplace law

 d B   = (1 / c). I. [d l ´ r ] / r 3   , (1.33a)

where d 1 is the element of the length of the conductor through which the stationary electric current I flows and r is the position vector directed from this current element to the examined point. From the Biot-Savart's law follows 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 pE / T whose divergence is always equal to zero, and that in the stationary case, coincides with a normal density "conductive" stream j. The expression jMaxw = (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

 Ief = nn j dS + nn (1/4p) ¶E/¶t dS .               conductive current   Maxwell displacement current (1.35)

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. jMaxw 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

 rot B   = (4p/c) j + (1/c) ¶E / ¶t . (1.34.b)

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 .

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

 div B   = 0    , (1.36b)

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 the existence of magnetic monopoles based on the idea of symmetry of electrodynamics equations. Experiments trying to find magnetic monopoles have not yet yielded any convincing results. However, magnetic monopolies are considered in modern quantum unitary field theories, which is related to their potential significance for the cosmology of the very early universe (Chapter 5, §5.5).

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 ?nC E dl induced along a closed curve C is proportional to the speed with which a change in magnetic flux F s nnSB 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 :

 rot E   = - (1/c) ¶B / ¶t . (1.37b)

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. These Maxwell's equations (1.31b) to (1.37b), which we gradually derived above, can be clearly summarized in differential form as follows :

 Maxwell's equations of the electromagnetic field (1.38) (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 . The classical physical system
is described by the integral of the action S

 S =   ò L (q1 , q2 , ... qn ,, q.1 , q.2 , ......... q.n ) d t ....

where L is the so-called Lagrange function - Lagrangian, describing all dynamic characteristics qi 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:

 L =   1/8p ( E 2 - B 2 ) + j . A - r . j . (1.42)

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 :

 D = e . E , B = m . H   ,

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 lower than in vacuum: c´ = 1/Ö(e.m) < c = 1/Ö(eo.mo) = 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

 E   = - grad j   - (1/c) ¶A / ¶t , (1.43a)
 B   =   rot A   . (1.43b)

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

 A ® A' = A + grad f ,   j ® j' = j - (1/c) ¶f/¶t , (1.44)

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 :

 grad A   + (1/c) ¶j / ¶ t = 0 ; (1.45)

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, jo and Ao 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: òòS f(...) dS and òòòV f(...) dV. In the following, however, for brevity we will use only one integration sign: òS f(...) dS and òV 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. In other words, the magnetic field can be considered a relativistic manifestationelectric 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, looking at the same system of charges, will see a current of charges, exciting a magnetic field - creating a magnetic field associated with the movement of charges - with an electric current - according to Biot-Savart-Laplace's law. The magnetic field appears as a "relativistic product" in Lorentz coordinate transformations in the presence of a quiescent electric field. However, all these connections are based on the relationship between electric and magnetic fields, expressed in Maxwell's equations.
This relativistic combination of electric and magnetic forces will be analyzed in more detail in §1.6, part "Four-imensional electrodynamics", where the electric and magnetic fields will be combined into a 4- tensor of electromagnetic fiel 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.
If we write Maxwell's equations (1.38) and (1.40) for the space domain, 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

 D E - (1/c2) ¶2E/¶t2 = 0 , D B - (1/c2) ¶2B/¶t2 = 0 (1.48)

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 xo at the time to will be in all places whose coordinates and time satisfy the equation x - xo = c.(t - to). It is thus a wave propagating in the direction of the X axis at the phase velocity c .
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".

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

 B   =   n ° x E   , (1.49)

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

 B = (1/c) (A. x n°) , B = (1/c) [(A. x n°) x n°] (1.49 ')

(the dot above A means the time derivative: A. s 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. = Ao(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 = Ao cos [w.(t - x/c) + a] ,

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

 k   = def    ( w / c). n° (1.50)

a plane wave can be expressed in form

 A(r,t)   =   Ao cos (wt - k.r + a) (1.51)

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

 A   =   Re [Âo . e i(k.r - wt)]   , (1.51 ')

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

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 ® Â' = eiJ. 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 field (electromagnetic waves and their quantum - photons) is therefore equal to s = 1 .

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

ee = (1/2) a=1SN qa.ja = (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 We = (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

em = (1/2) a=1SN Ia.Fa = (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 Wm = (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 :

 Welmag   = (1/8p) (E 2 + B 2 ) . (1.52)

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

 ¶ [(E2 + B2)/8] / ¶t  =   - div [(c/4p).(E ´ B)] - j . E   . (1.53)

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 eelmag 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 ekin 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

 P   = (c / 4p ). ( E x B ) (1.55)

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

 p   =    ò (1/4p c). ( E x B ) dV , (1.56)

so the momentum of the volume unit of the electromagnetic field is equal to P/c2 .

The energy current in a plane electromagnetic wave is equal to (1.49)

 P   =   (c/4p) ( E x B )   =   (c/4p) E2.n°   =   (c/4p) B2.n°   , (1.57)

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

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=1SNqava = (1/cR) (d/dt)a=1SNqar'a , i.e.

 A(R , t) = (1 / c.R). d. (t- R/c) , (1.58)

where d º S qa ra 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

 E(R,t) = (1/c2R) [ (d.. x R°) ´ R°] , B(R,t) = (1/c2R) (d.. x R°) , (1.59)

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)

 P = (c/4p) (E´B) = (1/4pc3R2) (d..xR°)2 = (d.. 2/4pc3R2) sin2J . R° , (1.60)

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

 I   º dE / dt  = (2.q2 / 3c3 ). a 2  . (1.61 ')

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 eo) 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 Kab = n r.(3xaxb - dab.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 ) (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

Ere  = (2 / 3c 3 ) d ...  .

In the source system, therefore, each charge q will be subjected to a certain additional force "reaction" fre = q. Ere acting per unit time of work fre.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 qava = (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 :

 m .v = q.E + (q/c) (v x B) + (2q2/3c3) v.. ; (1.63)

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 , , .

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 ai??? - aithér = heaven, 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.
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.
Note: In chemistry, aether is called a volatile organic solvent.

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.

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Vojtech Ullmann