Spreading electro magnetic field in space is a wave process, the description of which can be obtained from Maxwell's equations. Maxwell's equations describe the properties of electromagnetic waves in the most general case, but their direct use is not always convenient. Therefore, for the case of linear and homogeneous media, it is possible to obtain simpler wave equations, from which all the laws of geometric optics follow.
1.3.1. Wave equations
In optics, the change in electric and magnetic fields is often considered independently of each other, and then the vector nature of the field is not significant, and the electromagnetic field can be considered and described as scalar (like a sound field). The scalar theory is much simpler than the vector theory, and at the same time makes it possible to fairly deeply analyze the propagation of light beams and the processes of image formation in optical systems. In geometric optics, the scalar theory is widely used precisely because the electric and magnetic fields in this case can be described independently of each other, and the wave equations are the same for vector and scalar fields.
Let's consider the derivation of wave equations directly from Maxwell's equations. Let us take the equation for the rotor of the electric field, determined through the time derivative of the magnetic induction:
Vector multiply this equation by:
Considering that (1.5), we get:
Since the divergence of the electric field in a dielectric medium is , then in a homogeneous medium, which follows from Maxwell’s equations (4, 5). Then we get wave equation for the electric field component:
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Since, one vector equation splits into three scalar equations:
Arguing in a similar way, we can get wave equation for the magnetic component of the field:
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Since , then this vector equation also splits into three scalar equations:
From Maxwell's equations it follows that each of the components , , vector obeys absolutely the same scalar equation in form. Therefore, if we need to know the change in only one of the components of the vector, we can consider the vector field as a scalar one. Before finally moving on to the scalar theory, it should be noted that the components of the vector are not independent functions, which follows from the condition. Therefore, although scalar wave equations are a consequence of Maxwell’s equations, it is impossible to go back from them to Maxwell’s equations.
Let a scalar quantity be any of the components of the electric vector: ( , or ). In other words, this is a disturbance of the field at some point in space at some point in time. Then we can write wave equation in general:
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The second derivative of the disturbance with respect to time,
The meaning of this equation is that a wave is formed when a certain disturbance has a second derivative with respect to spatial coordinates that is proportional to the second derivative with respect to time.
It can be shown that the wave speed for dielectrics is related to the electric and magnetic constant of the medium as follows:
Consequently, the speed of wave propagation in space is determined as follows:
Then general form The wave equation can be written as follows:
Wave equation for one coordinate axis:
The ratio of the speed of light in a vacuum to the speed of light in a medium is called refractive index of a given medium relative to vacuum (index of refraction):
- cyclic frequency of field change over time,
- field phase (function of spatial coordinates).
Fig.1.3.1. Variation of the monochromatic field over time.
The monochromatic field is also characterized period of oscillation or frequency :
Moreover, the cyclic frequency can be expressed through frequency:
A harmonic wave is also characterized by a spatial period - wavelength :
AND wave number:
Radiation with a certain wavelength has a corresponding color (Fig. 1.3.2).
Fig.1.3.2. Visible radiation spectrum.
Constant characteristics, independent of the refractive index, for a monochromatic field are: frequency, cyclic frequency and oscillation period. Wavelength and wave number change depending on the refractive index, as the speed of light propagation in the medium changes. So, the frequency in the medium is always conserved, but the wavelength changes. The wavelength and wave number in a certain medium with a refractive index can be determined as follows:
Where is the wavelength in vacuum, is the wave number in vacuum.
Sometimes, when describing a monochromatic field, other concepts are used instead of phase. Let us introduce the wave number instead of the cyclic frequency into the expression for the wave disturbance:
Then the wave disturbance will be written as follows:
The word "eikonal" comes from Greek word(eikon - image). In Russian this corresponds to the word “icon”.
In contrast to the field phase, the eikonal is a more convenient quantity for assessing the phase change from ray to ray, since it is directly related to the geometric path length of the ray.
Optical beam length (optical path difference, OPD) is the product of the refractive index and the geometric path length.
The eikonal increment is equal to the optical beam length:
| (1.3.20) |
If the phase changes to , then the eikonal changes to: ;
if the phase changes to , then the eikonal changes to: ;
if the phase changes to , then the eikonal changes to: .
Eikonal has great value in the theory of optical imaging, since the concept of eikonal allows, firstly, to describe the entire process of image formation from the standpoint of the wave theory of light, and secondly, to most fully analyze the distortions in image transmission by optical instruments. The eikonal theory, developed in the 19th century by Petzval, Seidel and Schwarzschild, was an important fundamental achievement of geometric optics, thanks to which the creation of optical systems became possible High Quality. . When adding fields, their complex amplitudes are added, and the time exponential factor can be taken out of brackets and not taken into account:
1.3.4. Helmholtz equation
If the field is monochromatic, then differentiation with respect to time is reduced to multiplying the scalar amplitude by an imaginary factor. Thus, if we substitute the description of the monochromatic field (1.3.23) into the wave equation (1.3.18), then after transformations we will obtain a wave equation for the monochromatic field, which will include only the complex amplitude (Helmholtz equation).
Helmholtz equation(Helmgolz equation):
Maxwell's theory is based on the four equations considered:
1. The electric field can be either potential ( e q), and vortex ( E B), therefore the total field strength E=E Q+ E B. Since the circulation of the vector e q is equal to zero, and the circulation of the vector E B is determined by the expression, then the circulation of the total field strength vector 
This equation shows that the sources of the electric field can be not only electric charges, but also time-varying magnetic fields.
2. Generalized vector circulation theorem N:
This equation shows that magnetic fields can be excited either by moving charges or by alternating electric fields.
3. Gauss's theorem for the field D:
If the charge is distributed continuously inside a closed surface with volume density, then the formula will be written in the form 
4. Gauss's theorem for field B: 
So, the complete system of Maxwell's equations in integral form:
The quantities included in Maxwell’s equations are not independent and the following relationship exists between them: D= 0 E,
B= 0 N,j=E, where 0 and 0 are the electric and magnetic constants, respectively, and
- dielectric and magnetic permeability, respectively, - specific conductivity of the substance.
For stationary fields (E= const and IN=const) Maxwell's equations will take the form 
i.e., sources of electric field in in this case are only electric charges, sources of magnetic are only conduction currents. In this case, the electric and magnetic fields are independent of each other, which makes it possible to study separately permanent electric and magnetic fields.
IN 
Using the Stokes and Gauss theorems known from vector analysis, we can represent a complete system of Maxwell's equations in differential form:
Maxwell's equations are the most general equations for electric and magnetic fields in quiescent environments. They play the same role in the doctrine of electromagnetism as Newton's laws in mechanics. From Maxwell's equations it follows that an alternating magnetic field is always associated with the one generated by it electric field, and the variable electric field is always connected with the magnetic field generated by it, i.e. the electric and magnetic fields are inextricably linked with each other - they form a single electromagnetic field.
66. Differential equation of an electromagnetic wave. Plane electromagnetic waves.
For homogeneous And isotropic environment far from charges and currents, creating an electromagnetic field, it follows from Maxwell’s equations that the intensity vectors E And N alternating electromagnetic field satisfy the wave equation of the type:
- Laplace operator.
Those. electromagnetic fields can exist in the form of electromagnetic waves. The phase speed of electromagnetic waves is determined by the expression 
(1)
v
- phase velocity, where c = 1/ 0 0, 0 and 0 are the electric and magnetic constants, respectively, and are the electrical and magnetic permeabilities of the medium, respectively.
In vacuum (at =1 and =1) the speed of propagation of electromagnetic waves coincides with the speed With. Since > 1, the speed of propagation of electromagnetic waves in matter is always less than in vacuum.
When calculating the speed of propagation of the electromagnetic field using formula (1), a result is obtained that matches the experimental data quite well, if we take into account the dependence of and on frequency. The coincidence of the dimensional coefficient b with the speed of propagation of light in a vacuum indicates a deep connection between electromagnetic and optical phenomena, which allowed Maxwell to create the electromagnetic theory of light, according to which light is electromagnetic waves.
WITH 
a consequence of Maxwell's theory is the transverseness of electromagnetic waves: vectors E And N electric and magnetic field strengths of the wave are mutually perpendicular (Fig. 227) and lie in a plane perpendicular to the vector v of the speed of wave propagation, and the vectors E,
N And v form a right-handed system. From Maxwell’s equations it also follows that in an electromagnetic wave the vectors E And N always hesitate in the same phases(see Fig. 227), and the instantaneous values of £ and R at any point are related by the relation 0 = 0 N.(2)
E 
These equations are satisfied, in particular, by plane monochromatic electromagnetic waves(electromagnetic waves of one strictly defined frequency), described by the equations E at =E 0 cos(t-kx+), (3) H z =
H 0
cos(t-kx+),
(4), where e 0
And N 0
-
respectively, the amplitudes of the electric and magnetic field strengths of the wave, - the circular frequency of the wave, k=/v - wave number, - the initial phases of oscillations at points with the coordinate x= 0.
In equations (3) and (4), is the same, since the oscillations of the electric and magnetic vectors in an electromagnetic wave occur with the same phase.
Any oscillatory circuit emits energy. A changing electric field excites an alternating magnetic field in the surrounding space, and vice versa. Mathematical equations describing the relationship between magnetic and electric fields were derived by Maxwell and bear his name. Let us write Maxwell's equations in differential form for the case when there are no electric charges () and currents ( j= 0 ):
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The quantities and are the electric and magnetic constants, respectively, which are related to the speed of light in vacuum by the relation
Constant and characterize electrical and magnetic properties environment, which we will consider homogeneous and isotropic.
In the absence of charges and currents, the existence of static electric and magnetic fields is impossible. However, an alternating electric field excites a magnetic field, and vice versa, an alternating magnetic field creates an electric field. Therefore, there are solutions to Maxwell's equations in a vacuum, in the absence of charges and currents, where electric and magnetic fields are inextricably linked with each other. Maxwell's theory was the first to combine two fundamental interactions, previously considered independent. Therefore we are now talking about electromagnetic field.
The oscillatory process in the circuit is accompanied by a change in the field surrounding it. Changes occurring in the surrounding space propagate from point to point at a certain speed, that is, the oscillatory circuit emits electromagnetic field energy into the space surrounding it.
When the vectors and are strictly harmonic in time, the electromagnetic wave is called monochromatic.
Let us obtain from Maxwell's equations the wave equations for the vectors and .
Wave equation for electromagnetic waves
As noted in the previous part of the course, the rotor (rot) and divergence (div)- these are some differentiation operations performed by certain rules over vectors. Below we will take a closer look at them.
Let's take the rotor from both sides of the equation
In this case, we will use the formula proven in the mathematics course:
where is the Laplacian introduced above. The first term on the right side is zero due to another Maxwell equation:
As a result we get:
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Let's express rot B through an electric field using Maxwell's equation:
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and use this expression on the right side of (2.93). As a result, we arrive at the equation:
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Considering the connection
and entering refractive index environment
Let's write the equation for the electric field strength vector in the form:
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Comparing with (2.69), we are convinced that we have obtained the wave equation, where v- phase speed of light in the medium:
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Taking the rotor from both sides of Maxwell's equation
and acting in a similar way, we arrive at the wave equation for the magnetic field:
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The resulting wave equations for and mean that the electromagnetic field can exist in the form of electromagnetic waves, the phase velocity of which is equal to
In the absence of a medium (at ), the speed of electromagnetic waves coincides with the speed of light in vacuum.
Basic properties of electromagnetic waves
Let us consider a plane monochromatic electromagnetic wave propagating along the axis X:
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The possibility of the existence of such solutions follows from the obtained wave equations. However, the electric and magnetic field strengths are not independent of each other. The connection between them can be established by substituting solutions (2.99) into Maxwell’s equations. Differential operation rot, applied to some vector field A can be symbolically written as a determinant:
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Substituting here expressions (2.99), which depend only on the coordinate x, we find:
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Differentiating plane waves with respect to time gives:
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Then from Maxwell’s equations it follows:
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It follows, firstly, that the electric and magnetic fields oscillate in phase:
In other words, and in an isotropic environment,
Then you can choose the coordinate axes so that the vector is directed along the axis at(Fig. 2.27) :
Rice. 2.27. Oscillations of electric and magnetic fields in a plane electromagnetic wave
In this case, equations (2.103) take the form:
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It follows that the vector is directed along the axis z:
In other words, the electric and magnetic field vectors are orthogonal to each other and both are orthogonal to the direction of wave propagation. Taking this fact into account, equations (2.104) are further simplified:
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This leads to the usual relationship between wave vector, frequency and speed:
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as well as the connection between the amplitudes of field oscillations:
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Note that relationship (2.107) takes place not only for the maximum values (amplitudes) of the magnitudes of the electric and magnetic field strength vectors of the wave, but also for the current ones - at any time.
So, from Maxwell's equations it follows that electromagnetic waves propagate in a vacuum at the speed of light. At the time, this conclusion made a huge impression. It became clear that not only electricity and magnetism are different manifestations of the same interaction. All light phenomena, optics, also became the subject of the theory of electromagnetism. Differences in human perception of electromagnetic waves are related to their frequency or wavelength.
The electromagnetic wave scale is a continuous sequence of frequencies (and wavelengths) of electromagnetic radiation. Maxwell's theory of electromagnetic waves allows us to establish that in nature there are electromagnetic waves of various lengths, formed by various vibrators (sources). Depending on how electromagnetic waves are produced, they are divided into several frequency ranges (or wavelengths).
In Fig. Figure 2.28 shows the scale of electromagnetic waves.
Rice. 2.28. Electromagnetic wave scale
It can be seen that the wave ranges various types overlap each other. Therefore, waves of such lengths can be obtained different ways. There are no fundamental differences between them, since they are all electromagnetic waves generated by oscillating charged particles.
Maxwell's equations also lead to the conclusion that transversality electromagnetic waves in a vacuum (and in an isotropic medium): the electric and magnetic field strength vectors are orthogonal to each other and to the direction of wave propagation.
http://www.femto.com.ua/articles/part_1/0560.html – Wave equation. Material from the Physical Encyclopedia.
http://fvl.fizteh.ru/courses/ovchinkin3/ovchinkin3-10.html – Maxwell’s equations. Video lectures.
http://elementy.ru/trefil/24 – Maxwell’s equations. Material from "Elements".
http://nuclphys.sinp.msu.ru/enc/e092.htm – Very briefly about Maxwell’s equations.
http://telecomclub.org/?q=node/1750 – Maxwell’s equations and their physical meaning.
http://principact.ru/content/view/188/115/ – Briefly about Maxwell’s equations for the electromagnetic field.
Doppler effect for electromagnetic waves
Let in some inertial frame of reference TO A plane electromagnetic wave propagates. The wave phase has the form:
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Observer in another inertial frame TO", moving relative to the first one at a speed V along the axis x, also observes this wave, but uses different coordinates and time: t",r". The connection between reference systems is given by Lorentz transformations:
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Let's substitute these expressions into the expression for phase, to get the phase waves in a moving reference frame:
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This expression can be written as
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Where and - cyclic frequency and wave vector relative to the moving reference frame. Comparing with (2.110), we find the Lorentz transformations for frequency and wave vector:
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For electromagnetic wave in a vacuum
Let the direction of wave propagation make an angle with the axis in the first reference system X:
Then the expression for the frequency of the wave in the moving reference frame takes the form:
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That's what it is Doppler's formula for electromagnetic waves.
If , then the observer moves away from the radiation source and the wave frequency perceived by him decreases:
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If , then the observer approaches the source and the radiation frequency for it increases:
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At speeds V<< с we can neglect the deviation of the square root in the denominators from unity, and we arrive at formulas similar to formulas (2.85) for the Doppler effect in a sound wave.
Let us note an essential feature of the Doppler effect for an electromagnetic wave. The speed of the moving reference frame plays here the role of the relative speed of the observer and the source. The resulting formulas automatically satisfy Einstein's principle of relativity, and with the help of experiments it is impossible to establish what exactly is moving - the source or the observer. This is due to the fact that for electromagnetic waves there is no medium (ether) that would play the same role as air for a sound wave.
Note also that for electromagnetic waves we have transverse Doppler effect. When the radiation frequency changes:
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while for sound waves, movement in a direction orthogonal to the wave's propagation did not lead to a frequency shift. This effect is directly related to the relativistic time dilation in a moving frame of reference: an observer on a rocket sees an increase in the frequency of radiation or, in general, an acceleration of all processes occurring on Earth.
Let us now find the phase velocity of the wave
in a moving reference frame. From the Lorentz transformations for the wave vector we have:
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Let's substitute the ratio here:
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We get:
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From here we find the wave speed in the moving frame of reference:
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We found that the speed of the wave in the moving reference frame has not changed and is still equal to the speed of light With. Let us note, however, that, with correct calculations, this could not fail to happen, since the invariance of the speed of light (electromagnetic waves) in vacuum is the main postulate of the theory of relativity already “incorporated” into the Lorentz transformations we used for coordinates and time (3.109).
Example 1. Photon rocket moves at speed V = 0.9 s, heading for a star observed from Earth in the optical range (wavelength µm). Let's find the wavelength of radiation that the astronauts will observe.
The wavelength is inversely proportional to the vibration frequency. From formula (2.115) for the Doppler effect in the case of approaching the light source and the observer, we find the law of wavelength conversion:
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from which the result follows:
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According to Fig. 2.28 we determine that for the astronauts the star’s radiation has shifted to the ultraviolet range.
Energy and momentum of the electromagnetic field
Volumetric energy density w electromagnetic wave consists of volumetric densities of electric and magnetic fields.
1. Maxwell's equations and the wave equation. The electromagnetic field is described by Maxwell's equations: Consider a homogeneous and isotropic, electrically neutral, non-conducting medium.
1. Maxwell's equations and the wave equation. In the medium under consideration (ε = const. , μ = const. , = 0) these equations can be rewritten as follows: (1) (2) (3) (4) Let us calculate the rotor from the right and left sides of equation (1).
1. Maxwell's equations and the wave equation. According to equation (4) After calculating the rotor from the left side of equation (1), we obtain:
1. Maxwell's equations and the wave equation. Let's calculate the rotor from the right side of equation (1). According to equation (3) After calculating the rotor from the right and left sides of equation (1), we obtain:
1. Maxwell's equations and the wave equation. Let us compare the resulting equation with the general form of the differential wave equation: where v is the phase velocity of wave propagation. The equation we obtained for the electric field strength coincides with the wave equation if the solutions to the wave equation are plane waves of the form
1. Maxwell's equations and the wave equation. The solutions to the wave equation for the electric field strength vector are also plane waves. In this case, fluctuations in the electric field strength propagate in space. The phase speed of propagation in space of such oscillations is:
1. Maxwell's equations and the wave equation. Similarly, the wave equation can be derived by considering the magnetic field strength. In the medium under consideration (ε = const. , μ = const. , = 0): (1) (2) (3) (4) Let us calculate the rotor from the right and left sides of equation (3). Let us carry out the transformations, as in we use equation (2) and obtain: in the previous case,
1. Maxwell's equations and the wave equation. This equation can be rewritten as follows: where is the phase velocity of the wave. - solution of the wave equation, plane wave equation. Note that the solutions are the same for both the electric and magnetic fields. Fluctuations in the electric voltage and simultaneously occur in the magnetic field at the same speed. These oscillations are in phase. Fluctuations in the strength of electric and magnetic fields propagating in space are called electromagnetic waves.
1. Maxwell's equations and the wave equation. Phase velocity of an electromagnetic wave In a vacuum, when ε = 1 and μ = 1, In some medium, when ε > 1 and μ > 1, In optics, the quantity n is called the refractive index. Physical meaning refractive index - it shows how many times the speed of light (EMV) in a given medium is less than in vacuum.
1. Maxwell's equations and the wave equation. Main conclusions: 1. Maxwell's equations admit wave solutions. 2. The electromagnetic field represents fluctuations in the strength of electric and magnetic fields propagating in space. 3. The speed of propagation of electromagnetic waves in a vacuum 4. The speed of propagation of electromagnetic waves in any dielectric medium is less than in a vacuum: n is the refractive index of the medium.
2. Experimental discovery of electromagnetic waves. Scheme of Hertz's experiment. James Clark Maxwell (1831-1879) Heinrich Rudolf Hertz (1857 - 1894)
3. EMF cross-section. We have already noted some properties of electromagnetic waves: 1. The speed of propagation of electromagnetic waves in a vacuum 2. The speed of propagation of electromagnetic waves in any dielectric medium is less than in a vacuum: n is the refractive index of the medium. One more the most important property EMW is its transverseness.
3. EMF cross-section. If a plane electromagnetic wave propagates along the OX axis of the reference system we have chosen, then its equation can be written as follows: Here ω is the cyclic (circular) frequency of wave oscillations, k is the wave number. It is known that the wave surfaces of a plane wave are planes. If a wave propagates along the OX axis, then its wave surfaces are planes, parallel planes YZ (perpendicular to OX).
3. The cross-section of the electromagnetic wave propagates along the OX axis, the change in vectors E and H is described by the equations Each of the wave surfaces is characterized by one value of the X coordinate. Therefore, within the same wave surface at a given time, the values of the intensity vector are the same. This is true for both vector E and vector H. The values of all three components of vector E and all three components of vector H depend only on the X coordinate and do not depend on the Y and Z coordinates.
3. EMF cross-section. Let's consider the equation for the propagation of electromagnetic waves: On the left side of this equation The same for components: describing
3. EMF cross-section. In directions perpendicular to the direction of wave propagation, the time derivatives of H are not equal to zero; therefore, an alternating magnetic field can exist in these directions. In a direction parallel to the direction of wave propagation, only a stationary magnetic field can exist.
3. EMF cross-section. If we consider the equation describing the propagation of electromagnetic waves and, as in the previous case, rewrite it in the form of projections on the coordinate axes, and take into account that all components of the vector H depend only on the x coordinate, we obtain In directions perpendicular to the direction of propagation of the wave, there may be a variable electric field. In a direction parallel to the direction of wave propagation, only a stationary electric field can exist.
4. Electromagnetic wave polarization. If the oscillations of the electric field strength vector in a wave are somehow ordered, the wave is called polarized. If the oscillations of the electric field strength vector in a wave occur in one plane, the wave is called linearly polarized. If the plane in which the electric field strength vector oscillates in the wave rotates, the wave is called circularly polarized (elliptical).
5. The relationship between E and H in electromagnetic waves. Let's consider the equation describing the propagation of electromagnetic waves: On the left side of this equation
5. The relationship between E and H in electromagnetic waves. Let us take into account that the vector E depends only on the coordinate x. Consider the equation describing the propagation of electromagnetic waves: On the left side of this equation
5. The relationship between E and H in electromagnetic waves. Let us take into account that the vector H depends only on the coordinate x. The solutions to the wave equation are plane waves (the wave propagates along OX, the intensity vectors are perpendicular)
5. The relationship between E and H in electromagnetic waves. As we established earlier, let us substitute expressions for field strengths into this equation. This relationship must be satisfied at any time and at a point with any x coordinate.
5. The relationship between E and H in electromagnetic waves. The wave number k is related to the cyclic frequency ω by the relation
6. Umov-Poynting vector. It is known that the energy density of the electric field and the energy density of the magnetic field These expressions can be obtained from Maxwell's equations. Let's consider the equations: (1) (2) Let's multiply equation (1) by the vector H scalarly, and multiply equation (2) scalarly by the vector E.
6. Umov-Poynting vector. We similarly transform the second equation: We are considering a non-conducting medium, so j = 0. In total, we get two equations: Subtract the first from the second equation:
6. Umov-Poynting vector. Let us find out the physical meaning of the resulting expression. Let's denote the Umov-Poynting vector. - energy density of the electromagnetic field. Let's transform the left side of the equation:
6. Umov-Poynting vector. Let us apply the Ostrogradsky-Gauss theorem to the left side of the equation: Here is the surface surrounding the volume V. To ensure that the equality is not violated, we calculate the integral over the volume V and on the right side: Here Wem is the energy of the electromagnetic field in the volume V. Totally, it turns out:
6. Umov-Poynting vector. Thus, the flux of the Umov-Poynting vector through a certain closed surface is equal to the decrease in the energy of the electromagnetic field in the volume limited by this closed surface. According to the definition, Thus, These vectors form a right-handed triple. E and H lie in a plane perpendicular to the direction of wave propagation, the direction of S coincides with the direction of wave propagation.
7. Energy transferred by an electromagnetic wave. It is known that the energy density of the electromagnetic field If an electromagnetic wave propagates in space, then at a given point in space the energy density of the magnetic field At any time
7. Energy transferred by an electromagnetic wave. Let's introduce a new quantity, S, and call it the modulus of the energy flux density. That is, this value will be equal to the energy passing through a unit area per unit time W – energy, – area, t – time. Modulus of energy flux density (this value is equal to the energy passing through a unit area per unit time) equal to modulus Umov–Poynting vector.
7. Energy transferred by an electromagnetic wave. The energy of an electromagnetic wave passing through a unit area per unit time is equal to the modulus of the Umov–Poynting vector.
Maxwell's equations and the wave equation
Electromagnetic waves
In the process of distribution mechanical wave in an elastic medium oscillatory motion particles of the medium are involved. The reason for this process is the presence of interactions between molecules.
In addition to elastic waves, there is a wave process of a different nature in nature. We are talking about electromagnetic waves, which are the process of propagation of oscillations of the electromagnetic field. Essentially we live in a world of electromagnetic waves. Their range is incredibly wide - these are radio waves, infrared radiation, ultraviolet, x-rays, γ - rays. A special place in this diversity is occupied by the visible part of the range - light. It is with the help of these waves that we receive an overwhelming amount of information about the world around us.
What is an electromagnetic wave? What is its nature, mechanism of distribution, properties? Are there general patterns, characteristic of both elastic and electromagnetic waves?
Maxwell's equations and the wave equation
Electromagnetic waves are interesting because they were originally “discovered” by Maxwell on paper. Based on the system of equations he proposed, Maxwell showed that electric and magnetic fields can exist in the absence of charges and currents, propagating in the form of a wave with a speed of 3∙10 8 m/s. Almost 40 years later, the material object predicted by Maxwell—EMW—was discovered experimentally by Hertz.
Maxwell's equations are postulates of electrodynamics, formulated on the basis of an analysis of experimental facts. The equations establish the relationship between charges, currents and fields - electric and magnetic. Let's look at two equations.
1. Circulation of the electric field strength vector according to an arbitrary closed loop l is proportional to the rate of change of magnetic flux through a surface stretched over a contour (this is the law electromagnetic induction Faraday):
(1)
The physical meaning of this equation is that a changing magnetic field generates an electric field.
2. Circulation of the magnetic field strength vector along an arbitrary closed loop l is proportional to the rate of change in the flow of the electrical induction vector through the surface stretched over the contour:
The physical meaning of this equation is that the magnetic field is generated by currents and a changing electric field.
Even without any mathematical transformations of these equations, it is clear: if the electric field changes at some point, then in accordance with (2) a magnetic field appears. This magnetic field, changing, generates an electric field in accordance with (1). The fields mutually induce each other, they are no longer associated with charges and currents!
Moreover, the process of mutual induction of fields will propagate in space at a finite speed, that is, an electromagnetic wave appears. In order to prove the existence of a wave process in the system, in which the value S fluctuates, it is necessary to obtain the wave equation
Let us consider a homogeneous dielectric with dielectric constant ε and magnetic permeability μ. Let there be a magnetic field in this medium. For simplicity, we will assume that the magnetic field strength vector is located along the OY axis and depends only on the z coordinate and time t: .
We write equations (1) and (2) taking into account the relationship between the characteristics of fields in a homogeneous isotropic medium: and :
Let's find the vector flow through the rectangular area KLMN and the vector circulation along the rectangular contour KLPQ (KL = dz, LP= KQ = b, LM = KN = a)
It is obvious that the vector flux through the KLMN site and the circulation along the KLPQ circuit are different from zero. Then the circulation of the vector along the contour KLMN and the flux of the vector through the surface KLPQ are also non-zero. This is possible only under the condition that when the magnetic field changes, an electric field appears directed along the OX axis.
Conclusion 1: When the magnetic field changes, an electric field arises, the strength of which is perpendicular to the magnetic field induction.
Taking into account the above, the system of equations will be rewritten
After transformations we get:
