Is dispersion of electromagnetic waves observed in vacuum. Propagation of waves in dispersive media. Dispersion in a medium with free charges
Spatial and temporal dispersion of electromagnetic waves, group and phase velocity of waves in a medium with dispersion.
Wave phase velocity V f \u003d c / n in the general case, it may depend on the frequency (or the length of the radio wave 
, here 
is the refractive index of the medium. In this case, one speaks of the dispersion of the permittivity of the medium. Since the relationship between the Fourier components of the electric induction and electric field vectors is given by the relation 
, then the presence of dispersion means that the relative permittivity depends on the frequency or wave number. If is a function of frequency only 
, then one speaks of time dispersion if 
- about spatial.
The physical meaning of the time dispersion is as follows. Suppose that the elements of the environment (for example, electrons on the shell of atoms) under the influence electric field perform oscillations, the phase of which lags behind the phase of oscillations of the external wave. Then the waves emitted by these particles will experience an additional delay and arrive at the observation point later than the original electromagnetic wave. Spatial dispersion usually occurs if the length of an electromagnetic wave becomes comparable with the characteristic internal scales of the medium, which characterize the degree of influence of electromagnetic waves on its elements. Such scales can be the mean free path of particles, the radius of rotation of a charged particle in an external magnetic field (gyroradius), etc. In all the above cases, to determine the dispersion law, it is necessary to know the structure of the substance and the behavior of individual atoms or molecules in an external alternating electric field.
Consider a medium with dispersion in which the phase velocity V f = 
depends on the frequency of the wave 
. Any real wave, according to the Fourier theorem, can be represented as a sum of monochromatic waves with different amplitudes and frequencies. In a medium with dispersion, the propagation velocities of waves with different frequencies will be different. In the case when the frequency difference is much less than the average frequency, then such a wave packet is called narrow. Consider a superposition of two plane monochromatic waves of the same amplitude and close frequencies 
and 
, which correspond to the wave numbers 
and 
, propagating along the axis x
E(x, t) = E 0exp( 
+
E 0exp( 
Given the expression for the cosine of the angle cos 
=(exp ( i
)+exp(- i
)/2 following from the Euler formula exp
{i
)= cos 
+ i sin 
, we get
E(x,t)
= 2E 0 cos 
exp( 
}
This expression can be considered as an equation of a monochromatic wave, the amplitude of which varies depending on the spatial coordinate and time. The resulting signal is a beat with a slowly varying amplitude. The beat amplitude remains unchanged if 
=const . This means that the envelope of the wave packet propagates with the group velocity
.
The direction of the group velocity coincides with the direction of energy transfer by an electromagnetic wave. If the medium has no dispersion, then the group velocity coincides in magnitude with the phase velocity V gr = V f = 
and directed along 
.
Lecture 13. Maxwell's generalization of ideas about electromagnetic induction. Interrelation of variable electric and magnetic fields. Maxwell's equations in integral and differential forms, their physical interpretation Comparative characteristics electric and magnetic fields.
It is sometimes said about the classical theory of electromagnetic interaction and its carrier - the electromagnetic field - that Maxwell's electrodynamics are Maxwell's equations. In the 60s of the last century, Maxwell performed work similar to that which Newton had done two centuries before him. If Newton completed the creation of the first fundamental theory movements, then Maxwell completed the creation of the first theory of physical interactions(electromagnetic). Like Newton's classical mechanics, Maxwell's electrodynamics was also based on some extremely fundamental and elementary relations expressed by equations that received Maxwell's name.
These equations have two forms - integral and differential of their expression, and in fact they express the relationship of the characteristics of the electromagnetic field with the characteristics of the sources (charges and currents), this is the field of generators. This connection does not have such a simple expression as, for example, the connection between the measures of motion and interaction, expressed by the basic law of dynamics - Newton's second law. Therefore, Maxwell's equations, expressing the basic idea of electrodynamics - the doctrine of electromagnetic interaction - appear when studying it at a university - only at the end of the course.
Like any other extremely general theoretical propositions, Maxwell's equations are not formally derived within the framework of electrodynamics itself. They are obtained as a result of creative generalization of a variety of experimental material, and their correctness is confirmed by various consequences and practical applications.
Before Maxwell, the complete system of equations of electro- and magneto statics and one electro equation speakers- an equation expressing the law of electromagnetic induction. On the whole, this set of equations was not a complete system that unambiguously specifies the state of the electromagnetic field. To obtain such a system, Maxwell generalized the law of electromagnetic induction e = - dԤdt, writing his equation in integral form:
= -= - (the vector depends on both t and , and the flow Ф = - only on t)
The resulting equation can be thought of as a theorem on the circulation of a vector in electrostatics, generalized to a vortex electric field. Here, Maxwell actually threw out the conducting circuit that Faraday had and which, according to Maxwell, was simply an indicator of the presence (by induction currents) of an eddy electric field in the region around the changing magnetic field.
In the form of the law of electromagnetic induction presented by Maxwell, the physical essence of the phenomenon is more clearly visible, according to which an alternating magnetic field generates a vortex (with non-zero circulation) electric field in the surrounding space. Having presented the phenomenon of electromagnetic induction in this way, Maxwell was able, relying on symmetry considerations, to suggest the possibility of the existence in nature of the reverse electromagnetic induction effect. It can be called magnetoelectric induction, the essence of which is that a time-varying electric field generates a magnetic field in the surrounding space. Formally, this is written in such a way that the circulation of the magnetic field strength is equal to the rate of change in time of the electric field induction flux. Taking into account the fact that the magnetic field from the very beginning (from the static state) is vortex, that is, for it the circulation is always not equal to zero, the generalized relationship between the magnetic and electric fields will take the form:
I + I cm, where I cm =
Here, the rate of change of the electric field induction flux is formally equivalent to a certain current. This current is called bias current. It can be imagined that this current, as it were, closes the flow of current in a circuit, for example, with capacitors, through which the usual conduction current does not flow. The displacement current density is equal to the rate of change of the electric displacement (vector ): = (¶/¶t). When a charged capacitor is discharged, a conduction current flows through the wires, and, in addition, the electric field decreases (changes) in the space between the plates.
The speed of the change in the induction of the electric field, that is, ¶¤¶t, is the displacement current density. The displacement current closes the conduction current in the gaps between the conductors. It, like the conduction current, creates a magnetic field around itself, and in a dielectric (there it is called polarization current), it releases heat - the so-called dielectric losses.
So, now we can write down the complete system of equations of the unified electromagnetic field - the system of Maxwell's equations:
In a static state, an electric (electrostatic) field is generated only by stationary (or uniformly moving) electric charges in a given IFR and is potential (has zero circulation). The magnetostatic field is generated only by currents and is always non-potential (vortex). The electrostatic field, having charges as its sources, has the beginning of its lines of force on positive charges and the end - on negative charges (or at infinity). The magnetic field does not have such sources, since magnetic monopoles has not yet been discovered, and therefore lines of force even in a static state they are closed, having neither beginning nor end.
In a dynamic, non-stationary state, when the sources of fields and the fields themselves generated by them become time-varying, a new fundamental feature of the electric and magnetic non-stationary fields is revealed. It turns out that in this state they acquire the ability to give birth to each other, to become sources of each other. As a result, a new inextricably interconnected state of a single electromagnetic field. Maxwell's first equation, as already mentioned, indicates that a time-varying magnetic field generates a vortex electric field in the surrounding space. The second equation of Maxwell says that the magnetic field is generated not only by currents, but also by a time-varying electric field. As a result, we can conclude that variable (non-stationary) electric and magnetic fields are mutual sources of each other, and their difference is largely relative. In a non-stationary state, they are able to exist completely independently from the sources (alternating currents) that generated them, in the form of a single inseparable electromagnetic field.
The last two Maxwell's equations point to the different nature of the symmetry of the electric and magnetic stationary fields.
To solve the basic problem of electrodynamics, Maxwell's equations expressing its main idea (the relationship between the characteristics of the field and the characteristics of its sources) must be supplemented by the so-called material equations, linking the characteristics of the field with the characteristics of the real medium. These equations are as follows:
E about e; \u003d m about m and \u003d g, where e and m are the dielectric and magnetic permeability of the medium, and g is the electrical conductivity of the medium.
Maxwell's equations are often written in a more compact - differential form, which is obtained from the integral form by passing the contours and integration surfaces to the limit to zero: S ® 0 and L ® 0.
Let's introduce vector operator, called "nabla" and denoted Ñ , as a vector with the following components: Ñ = (¶/¶x, ¶/¶y, ¶/¶z).
For any vector field () = (A x, A y, A z), the following sets of differential operations are important:
a) scalar, called divergence:Ñ= diu = ¶A x /¶x + ¶A y /¶y + ¶A z /¶z
b) vector, called rotor :
Ñ = rot = (¶A y /¶ z - ¶A i /¶ y) + (¶A z /¶x - ¶A x /¶ z) + (¶A y /¶ X - ¶A X /¶ Y)
In these notations, Maxwell's equations in differential form take the following form:
rot= - ¶/¶t ; rot = + ¶/¶t; diu = r; diu = 0
or Ñ = -¶/¶t ; Ñ = + ¶/¶t; Ñ = r; Ñ = 0
Maxwell's equations only include free charges r and currents conductivity . Related charges and molecular currents enter these equations implicitly - through the characteristics of the medium - the dielectric and magnetic permeability e and m.
To pass to the differential form of writing the circulation theorem, we use the well-known Stokes theorem from vector analysis, which connects the circulation of a vector with the surface integral of the curl of this vector:
where S is the surface bounded by the contour L. The rotor of a vector is a vector differential operator defined as follows:
rot = (¶Е y /¶z - ¶Е z /¶у) + (¶E z /¶x - ¶E x /¶z) + (¶E x /¶y - ¶E y /¶x)
The physical meaning of the rotor is revealed by tending the surface S to zero. Within a sufficiently small surface, the rotor of the vector can be considered constant and taken out of the integral sign:
= rot × = rot×S.
Then, according to the Stokes theorem: rot = (1/S) as S ® 0.
From here vector rotor can be defined as surface circulation density of this vector.
Since the circulation of the vector in the ESP is zero, the rotor of the vector is also zero:
This equation is the differential form of the theorem on the circulation of a vector in an ESP.
To pass to the differential form of writing the Ostrogradsky-Gauss theorem, we use the Gauss theorem known from vector analysis, which connects the flow of a vector over a closed surface with the integral of the divergence of this vector over the volume contained in this surface:
The divergence of a vector is understood as a scalar differential operator (a set of derivatives) defined as follows:
div = ¶E x /¶x + ¶E y /¶y + ¶E z /¶z.
The physical meaning of the divergence is revealed by tending the volume V to zero. Within a sufficiently small volume, the divergence of the vector can be considered constant and taken out of the integral sign:
= div × = (1/V) div . Then, according to the Gauss theorem ,
div = (1/V) as V ® 0.
From here vector divergence can be defined as volumetric flux density of this vector.
Correlating the Ostrogradsky-Gauss theorem = q å /e o = (1/e o) and the Gauss theorem = , we see that their left parts are equal to each other. Equating their right sides, we get:
This equation is the differential form of the Ostrogradsky-Gauss theorem.
Lecture 14. Electromagnetic waves. Explanation of the emergence of electromagnetic waves from the standpoint of Maxwell's equations. The equation of a traveling electromagnetic wave. wave equation. Transfer of energy by an electromagnetic wave. Umov-Poynting vector. dipole radiation.
Electromagnetic waves are interconnected fluctuations of electric and magnetic fields propagating in space. Unlike sound (acoustic) waves, electromagnetic waves can propagate in a vacuum.
Qualitatively, the mechanism of the emergence of a free (from sources in the form of electric charges and currents) electromagnetic field can be explained on the basis of an analysis of the physical essence of Maxwell's equations. Two fundamental effects displayed by Maxwell's equations - electromagnetic induction(the generation of an alternating vortex electric field by an alternating magnetic field) and magnetoelectric induction(generation of an alternating electric field of an alternating magnetic field) lead to the possibility of electric and magnetic alternating fields to be mutual sources of each other. The interconnected change in electric and magnetic fields is a single electromagnetic field that can propagate in a vacuum at the speed of light
c \u003d 3 × 10 8 m / s. This field, which can exist completely independently of charges and currents and in general from matter, is the second (along with matter) - field type (form) of the existence of matter.
In the experiment, electromagnetic waves were discovered in 1886 by G. Hertz, 10 years after his death, who theoretically predicted their existence by Maxwell. From Maxwell's equations in a non-conductive medium, where r = 0 and = 0, taking the rotor operation from the first equation and substituting into it the expression for rot from the second equation , we get:
rot= - ¶/¶t = - m o m¶/¶t; rot rot= -m o m¶/¶t(rot) = - m o me o e¶ 2 /¶t 2 = - (1/u 2)¶E 2 /¶t 2 rot = ¶/¶t = e o e¶/¶t;
It is known from vector analysis that rot rot = grad div– D, but grad divº 0 and then
D= 1/u 2)¶ 2 /¶t 2 , where D = ¶ 2 /¶x 2 + ¶ 2 /¶y 2 + ¶ 2 /¶z 2 is the Laplace operator - the sum of second partial derivatives with respect to spatial coordinates.
In the one-dimensional case, we get differential equation in partial derivatives, called wave:
¶ 2 /¶x 2 - 1/u 2)¶ 2 /¶t 2 = 0
The same type of equation is obtained for the induction of a magnetic field. Its solution is a traveling plane monochromatic wave given by the equation:
Cos (wt - kx + j) and \u003d cos (wt - kx + j), where w / k \u003d u \u003d 1 /Ö (m o me o e) is the phase velocity of the wave.
The vectors and change in phase in time, but in mutually perpendicular planes and perpendicular to the direction of propagation (wave velocity): ^ , ^ , ^ .
The property of mutual perpendicularity of the vectors and and and allows us to attribute the electromagnetic wave to shear waves.
In a vacuum, an electromagnetic wave propagates at the speed of light u = c = 1/Ö(e o m o) = 3 × 10 8 m/s, and in a material medium the wave slows down, its speed decreases by a factor of Ö(em), that is, u = c/Ö(em) = 1/Ö(e o m o em).
At each point in space, the values of the vectors and are proportional to each other. The ratio of the strengths of the electric and magnetic fields is determined by the electric and magnetic properties(permeabilities e and m) of the medium. This expression is related to the equality of the volumetric energy densities w e and w m of the electric and magnetic fields of the wave:
w e \u003d e o eE 2 / 2 \u003d w m \u003d m o mH 2 / 2 Þ E / H \u003d Ö (m o m / e o e).
The ratio E / H, as it is easy to see, has the dimension of resistance: V / m: A / m \u003d V / A \u003d Ohm. In relation to vacuum, for example, E / H \u003d Ö (m o / e o) \u003d 377 Ohm - is called the vacuum impedance. The ratio E / B \u003d 1¤Ö (e o m o) \u003d c \u003d 3 × 10 8 m / s (in vacuum).
Spreading through space electromagnetic oscillations(electromagnetic waves) transfer energy without transferring matter - the energy of electric and magnetic fields. Previously, we obtained expressions for the volumetric energy densities of the electric and magnetic fields:
w e \u003d e about eE 2 / 2 and w m \u003d m about mH 2 ¤2 [J / m 3].
The main characteristic of energy transfer by a wave is the energy flux density vector, called (in relation to electromagnetic waves) the Poynting vector, numerically equal to the energy transferred through a unit area of the surface normal to the direction of wave propagation, per unit time: \u003d J / m 2 s \u003d W / m 2.
For a unit of time, all the energy that is contained in the volume V of a parallelepiped (cylinder) with a base of 1 m 2 and a height equal speed u of wave propagation, that is, the path traveled by the wave per unit time:
S = wV = wu = (w e + w m)¤Ö(e o m o em) = e o eE 2 ¤2Ö(e o m o em) + m o mH 2 ¤2Ö(e o m o em) = [Ö(e o e ¤m o m)]E 2 /2 + [Ö(m o m ¤e o e)] H 2 /2.
Since E / H \u003d Ö (m about m / e about e), then S \u003d EH / 2 + HE / 2 \u003d EH.
In vector form, the Poynting vector will be expressed as the product of the vectors of the electric and magnetic fields: = = w.
The simplest emitter of electromagnetic waves is an electric dipole, the moment of which changes over time. If the changes in the electric moment are repetitive, periodic, then such an "oscillating dipole" is called oscillator or basic vibrator. It represents the simplest (elementary) model of a radiative system in electrodynamics. Any electrically neutral radiator with dimensions L<< l в так называемой волновой или дальней зоне (при r >> l) has the same radiation field (character of distribution in space) as an oscillator with an equal dipole moment.
An oscillator is called linear or harmonic if its dipole moment changes according to the harmonic law: Р = Р m sin wt; R m = q l.
As radiation theory shows, the instantaneous power N of radiation of electromagnetic waves by a harmonic oscillator is proportional to the square of the second derivative of the change in its dipole moment, that is:
N ~ ïd 2 Р/dt 2 ï 2 ; N \u003d m o ïd 2 P / dt 2 ï 2 / 6pc \u003d m o w 4 R m 2 sin 2 wt / 6pc.
Average power< N >dipole radiation for the oscillation period is equal to:
< N >\u003d (1 / T) N dt \u003d m about w 4 R m 2 / 12pс
Noteworthy is the fourth power of frequency in the formula for the radiation power. In many ways, therefore, high-frequency carrier signals are used to transmit radio and television information.
The dipole radiates differently in different directions. In the wave (far) zone, the dipole radiation intensity J is: J ~ sin 2 q ¤r 2 , where q is the angle between the dipole axis and the direction of radiation. The dependence J (q) at a fixed r is called the polar radiation pattern of the dipole radiation. It looks like a figure eight. It can be seen from it that the dipole radiates most strongly in the direction q = p / 2, that is, in the plane perpendicular to the axis of the dipole. Along its own axis, that is, at q \u003d 0 or q \u003d p, the dipole does not radiate electromagnetic waves at all.
The equation of a traveling monochromatic wave Е = Е m cos (wt - kх + j) is an idealization of a real wave process. In fact, it must correspond to a sequence of humps and troughs, infinite in time and space, moving in the positive direction of the x axis with a speed u = w/k. This speed is called the phase speed, because it represents the speed of movement in space of the equiphase surface (constant phase surface). Indeed, the equation of the equiphase surface has the form
Real wave processes are limited in time, that is, they have a beginning and an end, and their amplitude changes. Their analytical expression can be represented as a set, group, wave package(monochromatic):
E \u003d E m w cos (wt - k w x + j w) dw
with close frequencies lying in a narrow interval from w - Dw/2 to w + Dw/2, where Dw<< w и близкими (не сильно различающимися) спектральными плотностями амплитуды Е м w , волновыми числами k w и начальными фазами j w .
When spread in a vacuum waves of any frequency have the same phase velocity u = c = 1¤Ö(e o m o) = 3×10 8 m/s, equal to the speed of light. AT material environment due to the interaction of an electromagnetic wave with charged particles (electrons, first of all), the wave propagation velocity begins to depend on the properties of the medium, its dielectric and magnetic permeability, according to the formula: u = 1/Ö(e o m o em).
The dielectric and magnetic permeability of a substance turn out to be dependent on the frequency (length) of an electromagnetic wave, and, consequently, the phase velocity of wave propagation in a substance turns out to be different for its different frequencies (wavelengths). This effect is called dispersion electromagnetic waves, and the media are called dispersive. A real medium can be non-dispersive only in a certain, not very wide frequency range. Only vacuum is a completely non-dispersive medium.
When propagating in a dispersive medium wave packet, its constituent waves with different frequencies will have different velocities and over time will "spread" relative to each other. The wave packet in such a medium will gradually blur, dissipate, which is reflected in the term "dispersion".
To characterize the propagation velocity of a wave packet as a whole, its propagation velocity is taken maximum- the center of the wave packet with the highest amplitude. This speed is called group and, in contrast to the phase velocity u = w/k, it is determined not in terms of the ratio w/k, but in terms of the derivative u = dw/dk.
Naturally, in a vacuum, that is, in the absence of dispersion, the phase velocity (speed of movement of the equiphase surface) and the group velocity (speed of energy transfer by a wave) coincide and are equal to the speed of light. The concept of group velocity, defined through the derivative (the rate of change of the angular frequency with increasing wave number) is applicable only for slightly dispersive media, where the absorption of electromagnetic waves is not very strong. We obtain the formula for the relationship between group and phase velocities:
u = dw/dk = u - (kl/k)×du/dl = u - l×du/dl.
Depending on the sign of the derivative du/dl, the group velocity u = u - l×du/dl can be either less or greater than the phase velocity u of the electromagnetic wave in the medium.
In the absence of dispersion, du/dl = 0, and the group velocity is equal to the phase velocity. With a positive derivative du/dl > 0, the group velocity is less than the phase velocity, we have a case called normal dispersion. With du/dl< 0, групповая скорость волн больше фазовой: u >u, this case of dispersion is called abnormal dispersion.
The causes and mechanism of the phenomenon of dispersion can be simply and clearly illustrated by the example of the passage of an electromagnetic wave through a dielectric medium. In it, an alternating electric field interacts with external electrons bound in the atoms of a substance. The strength of the electric field of an electromagnetic wave plays the role of a periodic driving force for an electron, imposing a forced oscillatory motion on it. As we have already analyzed, the amplitude of forced oscillations depends on the frequency of the driving force, and this is the reason for the dispersion of electromagnetic waves in a substance and the dependence of the permittivity of a substance on the frequency of an electromagnetic wave.
When the electron associated with the atom is displaced at a distance x from the equilibrium position, the atom acquires a dipole moment p = q e x, and the sample as a whole is a macrodipole with polarization P = np = nq e x, where n is the number of atoms per unit volume , q e is the electron charge.
From the connection of the vectors and one can express the dielectric susceptibility a, the permeability e, and then the speed u of an electromagnetic wave in a substance:
P \u003d e o aE \u003d nq e x Þ a \u003d nq e x / e o E; e \u003d 1 + a \u003d 1 + nq e x / e o E; u = s/Ö(em) » s/Öe (for m » 1). For small x: u = c/Ö(1 + nq e x/e o E) » c/(1 + nq e x/2e o E).
Based on Newton's second law for an electron elastically bound to an atom and located in a perturbing electric field E = E m cos wt of an electromagnetic wave, we find its displacement x from the equilibrium position in the atom. We believe that the displacement x of the electron changes according to the law of the driving force, that is, x \u003d X m cos wt.
ma = - kx - ru + F out; mx ¢¢ \u003d - kx - rx ¢ + q e E, or, with r \u003d 0 Þ x ¢¢ + w about 2 x \u003d q e E m cos wt / m,
where w o 2 = k/m is the natural oscillation frequency of an electron elastically bound to an atom.
We substitute the solution x = X m cos wt into the obtained differential equation of forced oscillations of an electron:
W 2 x + w o 2 x \u003d q e E m cos wt / m Þ x \u003d q e E m cos wt / \u003d q e E /
We substitute the resulting expression for the displacement x into the formula for the phase velocity of an electromagnetic wave:
u » c/(1 + nq e x/2e o E) = c/
At the frequency w = w o the phase velocity u of the electromagnetic wave vanishes.
At a certain frequency w p, at which nq e 2 /me o (w o 2 - w p 2) = - 1, the phase velocity of the wave undergoes a discontinuity. The value of this "resonant" frequency is w p \u003d w o + nq e 2 / me o "10 17 s -1.
Let us depict the obtained dependence of the phase velocity on the frequency and on the wavelength. The discontinuous nature of the dependence u(w), called dispersion, is due to the fact that we neglected the resistance of the medium and the dissipation of the vibrational energy, setting the drag coefficient r = 0. Accounting for friction leads to smoothing of the dispersion curve and elimination of discontinuities.
Since the frequency w and the wavelength l are inversely proportional (w = 2pn = 2pс/l), the plot of the dispersion dependence u(l) is inverse to the plot of u(w).
In the area of normal dispersion 1 - 2, the phase velocity u is greater than the speed of light in vacuum. This does not contradict the theory of relativity, because a real signal (information, energy) is transmitted with a group velocity u, which here is less than the speed of light.
The group velocity u = u - l×du/dl exceeds the speed of light c in vacuum in the anomalous dispersion region 2 – 3, where the phase velocity u decreases with increasing wavelength l and the derivative du/dl< 0. Но в области аномальной дисперсии имеет место сильное поглощение, и понятие групповой скорости становится неприменимым.
Lecture 16. Concepts of space and time in modern physics. Unification of space with time in SRT. Relativity of classical concepts of simultaneity, length and duration.
In 1905, A. Einstein for the first time issued in theoretical system kinematic, i.e. space-time representations, "prompted" by the experience of analyzing motions with large, so-called relativistic (commensurate with the speed of light c = 3 × 10 8 m / s in vacuum) velocities.
In Newton's mechanics, space-time representations were not specifically singled out and were actually considered obvious, consistent with the visual experience of slow motions. However, attempts made in the 19th century to explain, on the basis of these ideas, the features of the propagation of such a relativistic object as light, led to a contradiction with experience (experiment of Michelson, 1881, 1887, etc.). Analyzing the emerging problem situation, A. Einstein managed in 1905 to formulate two fundamental statements, called postulates (principles), consistent with the experience of relativistic (high-speed) motions. These statements, called Einstein's postulates, formed the basis of his special (private) theory of relativity.
1. Einstein's principle of relativity: all laws of physics are invariant with respect to the choice of inertial reference frame (ISR), i.e. in any IFR, the laws of physics have the same form, do not depend on the arbitrariness of the subject (scientist) in choosing the IFR. Or, in other words, all ISOs are equal, there is no privileged, elected, absolute ISO. Or, moreover, no physical experiments carried out inside the ISO can determine whether it is moving at a constant speed or at rest. This principle is consistent with the principle of objectivity of knowledge.
Before Einstein, the principle of relativity of Galileo was known in mechanics, which was limited to the framework of only mechanical phenomena and laws. Einstein actually generalized it to any physical phenomena and laws.
2. The principle of invariance (constancy) and limiting the speed of light. The speed of light in vacuum is finite, the same in all IFRs, i.e., it does not depend on the relative motion of the light source and receiver, and is the limiting speed of transmission of interactions. This principle consolidated in physics the concept of short-range interaction, which replaced the previously dominant concept of long-range interaction, based on the hypothesis of the instantaneous transmission of interactions.
From the two principles (postulates) of Einstein follow the most important for kinematics, more general than the classical (Galilean) transformations, that is, the formulas for the relationship of spatial and temporal coordinates x, y, z, t of the same event observed from different IFRs.
Let us take a special case of choosing two IFRs, in which one of them, denoted by (K), moves relative to the other, denoted by (K ¢), with a speed V along the x axis. At the initial moment of time, the origins of coordinates O and O ¢ of both IFRs coincided, and the axes Y and Y ¢ , as well as Z and Z ¢ , also coincided. For this case, the transformation formulas for the space-time coordinates of the same event in the transition from one IFR to another, called Lorentz transformations, have the following form:
x ¢ \u003d (x - Vt) / Ö (1 - V 2 / s 2); y ¢ = y; z ¢ = z; t ¢ \u003d (t - Vx / s 2) / Ö (1 - V 2 / s 2) -
Direct Lorentz transformations (from ISO (K) to ISO (K ¢);
x \u003d (x ¢ + Vt ¢) / Ö (1 - V 2 / s 2); y = y ¢; z = z ¢ ; t \u003d (t ¢ + Vx ¢) / Ö (1 - V 2 / s 2) -
Inverse Lorentz transformations (from ISO (K ¢) to ISO (K).
The Lorentz transformations are more general than the Galilean transformations, which they contain as a special, limiting case, valid at low, pre-relativistic velocities (u<< с и V << с) движений тел и ИСО. При таких, «классических» скоростях, Ö(1 – V 2 /с 2) » 1, и преобразования Лоренца переходят в преобразования Галилея:
x ¢ \u003d x - Vt; y ¢ = y; z ¢ = z; t ¢ \u003d t and x \u003d x ¢ + Vt ¢; y = y ¢; z = z ¢ ; t = t¢
In such a correlation of Lorentz's and Galileo's transformation formulas, an important methodological principle of scientific and theoretical knowledge, the principle of correspondence, finds its manifestation. According to the principle of correspondence, scientific theories develop dialectically along the path of stepwise generalization - expansion of their subject area. At the same time, a more general theory does not cancel the former, particular one, but only reveals its limitations, outlines the boundaries and limits of its justice and applicability, and itself reduces to it in the area of these boundaries.
The term "special" in the name of Einstein's theory of relativity means just that it is itself limited (particular) in relation to another theory, also created by A. Einstein, called "general relativity". It generalizes the special theory of relativity to any, not only inertial frames of reference.
A number of kinematic consequences follow from the Lorentz transformations, which contradict visual classical concepts and give grounds to call relativistic kinematics and relativistic mechanics as a whole the theory of relativity.
What about, that is, depending on the choice of ISO in SRT? First of all, the fact of the simultaneity of two events, as well as the length of the body and the duration of the process, turns out to be relative. In the relativistic dynamics strength passes into the category of relative ones, and for some scientists even mass. However, it should be remembered that the main thing in any theory is not the relative, but the invariant (stable, conserved, unchanging). Relativistic mechanics, revealing the relativity of some concepts and quantities, replaces them with other invariant quantities, such as, for example, a combination (tensor) of energy-momentum.
1. Relativity of the simultaneity of events.
Let two events occur in the IFR (K), given by the coordinates x 1, y 1, z 1, t 1 and x 2, y 2, z 2, t 2, and t 1 = t 2, i.e. in the IFR ( C) these events happen at the same time.
Einstein's great merit was to draw attention to the fact that in the classical mechanics of Galileo - Newton it was not at all determined how to fix the fact of the simultaneity of two events located in different places. Intuitively, in accordance with the principle of long-range action, which assumes an infinite speed of propagation of interactions (which is quite justified for slow motions), it was considered obvious that the spacing of events in space cannot affect the nature of their time relationship. Einstein proposed a rigorous way to establish the fact of simultaneity different places events based on the placement of synchronized clocks in those locations. He proposed to synchronize the clock with the help of a real signal with the highest speed - a light signal. One of the ways to synchronize clocks in a particular ISO is as follows: a clock located at a point with coordinate x will be synchronized with a single center at point 0 - the beginning of ISO, if at the moment a light signal emitted from point 0 at time t o arrives at them, they show the time t x \u003d t o + x / c.
Since synchronization is carried out by a signal that has an extremely high, but not infinite, speed, the clocks synchronized in one IFR will be out of sync in another (and in all other) IFRs due to their relative movement. The consequence of this is the relativity of the simultaneity of events of different places and the relativity of time and space intervals (durations and lengths).
Formally, this conclusion follows from the Lorentz transformations as follows:
in ISO (K ¢) event 1 corresponds to time t 1 ¢ = (t 1 - Vx 1 / s 2) / Ö (1 - V 2 / s 2), and event 2 ® corresponds to time t 2 ¢ = (t 2 - Vx 2 / s 2) / Ö (1 - V 2 / s 2), so that at t 1 \u003d t 2, t 2 ¢ - t 1 ¢ \u003d [(x 1 - x 2) V / s 2] / Ö(1 - V 2 /s 2), and two events 1 and 2, simultaneous in one IFR - in IFR (K), turn out to be non-simultaneous in another (in IFR (K ¢).
In the classical (pre-relativistic) limit, for V << с, t 2 ¢ – t 1 ¢ » 0, the fact of the simultaneity of two events becomes absolute, which, as already mentioned, corresponds to an infinite transmission rate of interactions and a synchronizing signal: с ® ¥ or с >> V.
In the relativistic theory, the simultaneity of events is absolute only
in the special case of single events: at x 1 = x 2 always at t 1 = t 2 and t 1 ¢ = t 2 ¢.
2. Relativity of the length of bodies (spatial intervals).
Let a rod of length l o \u003d x 2 - x 1.
IFR, in which the body is at rest, is called proper for this body, and its characteristics, in this case, the length of the rod, are also called proper.
In ISO (K ¢), relative to which the rod moves, and which is called the laboratory ISO, the length of the rod l¢ \u003d x 2 ¢ - x 1 ¢ is defined as the difference in the coordinates of the ends of the rod, fixed simultaneously by the clock of a given ISO, i.e., at t 1 ¢ = t 2 ¢.
Using the Lorentz transformation formulas for x 1 and x 2 containing time in the hatched ISO (K ¢), we establish the relationship l and l ¢ :
x 1 = (x 1 ¢ + Vt 1 ¢) / Ö (1 - V 2 / s 2); x 2 \u003d (x 2 ¢ + Vt 2 ¢) / Ö (1 - V 2 / s 2); Þ x 2 - x 1 \u003d (x 2 ¢ - x 1 ¢) / Ö (1 - V 2 / s 2)
or finally: l ¢ = l o Ö (1 - V 2 / s 2) - this formula expresses the law of length conversion
(spatial intervals), according to which the dimensions of the bodies are reduced in the direction of movement. This effect of the relativity of the length of bodies, their relativistic contraction in the direction of movement, is a real and not an apparent physical effect, but not dynamic, not associated with any force action that causes compression of the bodies and reduction in their size. This effect is purely kinematic, associated with the chosen method for determining (measuring) the length and the finiteness of the propagation velocity of interactions. It can also be explained in such a way that the concept of length in SRT ceased to be a characteristic of only one body, by itself, but became a joint characteristic of the body and the frame of reference (like the speed of a body, its momentum, kinetic energy, etc.).
Such characteristics change for different bodies in the same ISO, which is natural and familiar to us. But in the same way, although less familiar, they also change for the same body, but in different ISOs. At low speeds, this effect of the dependence of the body length on the choice of ISO is practically imperceptible, which is why it did not attract attention in Newton's mechanics (mechanics of slow motions).
A similar analysis of the Lorentz transformations in order to clarify the relationship between the durations of two processes measured from different IFRs, one of which is its own, i.e. e. moves along with the carrier of the process and measures its duration (the difference between the moments of the end and the beginning of the process) about the same clock, leads to the following results:
\u003d o (1 - V 2 s 2), where o is the own duration of the process (counted by the same clock moving along with the events taking place, and - the duration of the same process, counted by different clocks in ISO, relative to which the carrier of the process moves and at the moments of the beginning and end of the process it is in its different places.
Sometimes this effect is interpreted as follows: they say that a moving clock runs slower than a stationary one, and from this they derive a number of paradoxes, in particular the paradox of twins. It should be noted that due to the equality of all IFRs in SRT, all kinematic effects (both length reduction in the direction of motion and time dilation - duration by clocks moving relative to the carrier of the process) are reversible. And a good example of this reversibility is the experience with muons, unstable particles formed as a result of interaction with the atmosphere, bombarding it with cosmic rays. Physicists were initially surprised by the existence of these particles at sea level, where they would have to decay during their lifetime, i.e., not have time to fly from the upper layers of the atmosphere (where they are formed) to sea level.
But the point turned out to be that physicists first used in their calculations the intrinsic lifetime of -mesons o = 210 -6 s, and the distance they traveled was taken as a laboratory one, that is
l = 20 km. But either in this case it is necessary to take the length (the path traveled by -mesons) as well, which turns out to be "reduced", "shortened" according to the factor (l –V 2 /s 2). Or you need not only the length, but also the time to take the laboratory, and it increases in proportion to 1 / (l–V 2 / s 2). Thus, the relativistic effects of the transformation of time and space intervals allowed physicists to make ends meet in a real experiment and a natural phenomenon.
At low speeds V with the relativistic formula for the transformation of the durations of processes turns into the classical one . Accordingly, the duration in this limiting case (approximation) loses its relativistic relativity and becomes absolute, i.e., independent of the choice of ISO.
Revised in SRT and the law of addition of velocities. Its relativistic (general) form can be obtained by taking the differentials from the expressions for x, x , t and t , in the Lorentz transformation formulas and dividing dx by dt and dx by dt , that is, by forming speeds from them
x = dх/dt and x = dх /dt .
dx \u003d (dx + Vdt ) / (l -V 2 / s 2); dt \u003d (dt + Vdx / s 2) / (l -V 2 / s 2);
dх/dt = (dх + Vdt )/(dt + Vdх /с 2) = (dх /dt + V)/ x = ( x + V)(1 + V x / s 2)
dx \u003d (dx - Vdt) / (l -V 2 / s 2); dt \u003d (dt - Vdx / s 2) / (l -V 2 / s 2);
dx / dt = (dx - Vdt) / (dt - Vdx / s 2) = (dx / dt - V) / x = ( x - V) (1 - V x / s 2 )
Formulas x = ( x + V)(1 + V x /s 2) and x = ( x - V)(1 - V x /s 2) and express
relativistic laws of addition of velocities or, in other words, the transformation of velocities
when moving from ISO (K) to ISO (K ) and vice versa.
In the pre-relativistic limit of low speeds c these formulas turn into well-known expressions of the classical (Galilean) law of addition of velocities: x = x + V and x = x – V.
It is interesting to see how the relativistic form of the law of addition of velocities is consistent with the principle of constancy of the speed of light in all IFRs. If in IFR (K ) we have the speed x = c and IFR (K ) moves relative to IFR (K) also at a speed V = c, then relative to IFR (K) the speed of light will still be equal to c:
x \u003d ( x + V) (1 + V x / s 2) \u003d (s + s) (1 + s s / s 2) \u003d s. The classical law of addition led to the result: x = x + V = c + c = 2c, i.e., it contradicted experience, because it did not contain
in itself restrictions on the "ceiling" of speeds.
It follows from Maxwell's macroscopic electromagnetic theory that the absolute refractive index of the medium
where is the dielectric constant of the medium, is the magnetic permeability. In the optical region of the spectrum for all substances 1, therefore
From this formula, some contradictions with experience are revealed: the value n, being a variable, remains at the same time equal to a certain constant - . In addition, the values of n obtained from this expression do not agree with the experimental values. Difficulties in explaining the dispersion of light from the point of view of Maxwell's electromagnetic theory are eliminated by Lorentz's electron theory. In the Lorentz theory, the dispersion of light is considered as the result of the interaction of electromagnetic waves with charged particles that are part of the substance and perform forced oscillations in the alternating electromagnetic field of the wave.
Let us apply the electronic theory of light dispersion for a homogeneous dielectric, assuming formally that the dispersion of light is a consequence of the dependence on the frequency of light waves. The permittivity of a substance is
where w is the dielectric susceptibility of the medium, 0 is the electrical constant, P is the instantaneous value of the polarization. Hence,
i.e., it depends on R. In this case, the electron polarization is of primary importance, i.e., forced oscillations of electrons under the action of the electric component of the wave field, since for the orientational polarization of molecules, the frequency of oscillations in a light wave is very high (v 10 15 Hz) .
In the first approximation, we can assume that forced oscillations are performed only by external electrons, which are most weakly connected with the nucleus - optical electrons. For simplicity, let us consider oscillations of only one optical electron. The induced dipole moment of an electron performing forced oscillations is p = ex, where e is the charge of the electron, x is the displacement of the electron under the action of the electric field of the light wave. If the concentration of atoms in the dielectric is n 0 then the instantaneous value of the polarization
Consequently, the problem is reduced to determining the displacement x of an electron under the action of an external field E. The field of a light wave will be considered a function of the frequency co, i.e., changing according to the harmonic law: E = E 0 cost.
The equation of forced oscillations of an electron for the simplest case (without taking into account the resistance force that determines the absorption of the energy of the incident wave) can be written as
where F 0 = eE 0 is the amplitude value of the force acting on the electron from the wave field, is the natural oscillation frequency of the electron, m is the mass of the electron. Having solved the equation, we find = n 2 depending on the constants of the atom (e, m, 0) and the frequency of the external field, i.e. we will solve the dispersion problem. The solution of the equation can be written as
If there are different charges eh in the substance that perform forced oscillations with different natural frequencies ea0|, then
where m 1 is the mass of the i-th charge.
It follows from the expressions and that the refractive index n depends on the frequency of the external field, i.e., the dependences obtained really confirm the phenomenon of light dispersion, although under the above assumptions, which must be eliminated in the future. It follows from the expressions and that in the region from = 0 to = 0 n 2 is greater than one and increases with increasing (normal dispersion); at = 0 n 2 = ± ; in the region from = 0 to = n 2 is less than one and increases from - to 1 (normal dispersion). Passing from n 2 to n, we obtain that the dependence of n on has the form shown in Fig. 3.
This behavior of n near 0 is the result of the assumption that there are no resistance forces during electron oscillations. If this circumstance is also taken into account, then the graph of the function l(co) near too is given by the dashed line AB. The AB region is the region of anomalous dispersion (n decreases with increasing), the remaining sections of the dependence of n on describe normal dispersion (n increases with increasing).
Russian physicist D.S. Rozhdestvensky (1876-1940) belongs to the classic work on the study of anomalous dispersion in sodium vapor. He developed an interference method for very accurate measurement of the refractive index of vapors and experimentally showed that the formula
correctly characterizes the dependence of n on, and also introduced a correction into it that takes into account the quantum properties of light and atoms.
Today, quantitative knowledge of the electronic structure of atoms and molecules, as well as solids built from them, is based on experimental studies of optical reflection, absorption, and transmission spectra and their quantum mechanical interpretation. The band structure and defectiveness of various types of solids (semiconductors, metals, ionic and atomic crystals, amorphous materials) are being studied very intensively. Comparison of the data obtained in the course of these studies with theoretical calculations made it possible to reliably determine for a number of substances the features of the structure of energy bands and the values of interband gaps (band gap E g) in the vicinity of the main points and directions of the first Brillouin zone. These results, in turn, make it possible to reliably interpret such macroscopic properties of solids as electrical conductivity and its temperature dependence, the refractive index and its dispersion, the color of crystals, glasses, ceramics, glass-ceramics and its variation under radiation and thermal effects.
2.4.2.1. Dispersion of electromagnetic waves, refractive index
Dispersion is a phenomenon of the relationship between the refractive index of a substance, and, consequently, the phase velocity of wave propagation, with the wavelength (or frequency) of radiation. Thus, the transmission of visible light through a glass trihedral prism is accompanied by decomposition into a spectrum, with the violet short-wavelength part of the radiation deviated most strongly (Fig. 2.4.2).
The dispersion is called normal if, as the frequency n(w) increases, the refractive index n also increases dn/dn>0 (or dn/dl<0). Такой характер зависимости n от n наблюдается в тех областях спектра, где среда прозрачна для излучения. Например, силикатное стекло прозрачно для видимого света и обладает в этом интервале частот нормальной дисперсией.
The dispersion is called anomalous if, with increasing radiation frequency, the refractive index of the medium decreases (dn/dn<0 или dn/dl>0). Anomalous dispersion corresponds to frequencies corresponding to optical absorption bands; the physical content of the absorption phenomenon will be briefly discussed below. For example, for sodium silicate glass, the absorption bands correspond to the ultraviolet and infrared regions of the spectrum, quartz glass in the ultraviolet and visible parts of the spectrum has normal dispersion, and in the infrared - anomalous.
Rice. 2.4.2. Dispersion of light in glass: a - decomposition of light by a glass prism, b - graphs n = n (n) and n = n (l 0) for normal dispersion, c - in the presence of normal and anomalous dispersion In the visible and infrared parts of the spectrum, normal dispersion is characteristic for many alkali-halide crystals, which determines their wide use in optical devices for the infrared part of the spectrum.
The physical nature of normal and anomalous dispersion of electromagnetic waves becomes clear if we consider this phenomenon from the standpoint of classical electron theory. Let us consider a simple case of normal incidence of a plane electromagnetic wave of the optical range on a flat boundary of a homogeneous dielectric. The electrons of a substance associated with atoms under the action of an alternating field of a wave with strength 
perform forced oscillations with the same circular frequency w, but with a phase j that differs from the phase of the waves. Taking into account the possible attenuation of the wave in a medium with a natural frequency of electron oscillations w 0 , the equation of forced transverse oscillations in the direction - the direction of propagation of a plane polarized wave - has the form
(2.4.13)
known from the course of general physics (q and m - the charge and mass of the electron).
For the optical region, w 0 » 10 15 s -1 , and the attenuation coefficient g can be determined in an ideal medium under the condition of a non-relativistic electron velocity (u< At w 0 = 10 15 s -1 the value g » 10 7 s -1 . Neglecting the relatively short stage of unsteady oscillations, let us consider a particular solution of the inhomogeneous equation (2.4.13) at the stage of steady oscillations. We are looking for a solution in the form Then from equation (2.4.13) we obtain or here Then the solution for the coordinate (2.4.15) can be rewritten as Thus, the forced harmonic oscillations of an electron occur with amplitude A and are ahead in phase of the oscillations in the incident wave by an angle j. Near the resonance value w = w 0 , the dependence of A and j on w/w 0 is of particular interest. Rice. 2.4.3. Graphs of the amplitude (a) and phase (b) of electron oscillations near the resonant frequency (for g » 0.1w 0) In real cases, usually g is less than g » 0.1 w 0 , chosen for clarity in Fig. 2.4.3, the amplitude and phase change more sharply. If the light incident on the dielectric is not monochromatic, then near the resonance, at frequencies w®w 0, it is absorbed, the electrons of the substance dissipate this energy in the volume. This is how absorption bands appear in the spectra. The line width of the absorption spectrum is determined by the formula
(2.4.14)
(2.4.15)
, where the oscillation amplitude is equal to
(2.4.16)
(2.4.17)
On fig. 2.4.3 shows the graphs of the dependences of the amplitude and phase near the resonant frequency.
