Maxwell's equations and the wave equation

Electromagnetic waves

During the propagation of a mechanical wave in an elastic medium, particles of the medium are involved in oscillatory motion. 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 that are 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 along an arbitrary closed loop l is proportional to the rate of change of magnetic flux through a surface stretched over a contour (this is Faraday’s law of electromagnetic induction):

(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:

Maxwell's equations contain a continuity equation expressing the law of conservation of charge. 3. Maxwell's equations are satisfied in all inertial systems of the report. 4. Maxwell's equations are symmetrical.

6.3.4. Electromagnetic waves

From Maxwell's equations it follows that the electromagnetic field is capable of existing independently, without electric charges and currents. The changing electromagnetic field has a wave character and propagates in a vacuum in the form of electromagnetic waves at the speed of light.

The existence of electromagnetic waves follows from Maxwell’s equations, which are described by wave equations for vectors and respectively:

, (5.18)

, (5.19)

A change in time of a magnetic field excites an alternating electric field and, conversely, a change in time of an electric field excites an alternating magnetic field. Vortex electric field induced by alternating magnetic field , forms with the vector left-handed system (Fig. 7.2), and the vortex magnetic field induced by the electric field , forms with the vector right-handed screw system (Fig. 5.2).

Their continuous interconversion occurs, which makes it possible

exist and spread in space and time in the absence of charges and currents.

Thus, Maxwell’s theory not only predicted the existence of electromagnetic waves, but also established their most important properties:

Speed of propagation of an electromagnetic wave in a neutral non-conducting and non-ferromagnetic medium

(5.20)

where c is the speed of light in vacuum.

Rice. 5.3 Fig. 5.4

3. In an electromagnetic wave, vectors

And

always oscillate in the same phases (Fig. 5.4), and between the instantaneous values of E and B at any point in space

there is a connection, namely: E = vB or
. (5.21)

The existence of electromagnetic waves allowed Maxwell to explain the wave nature of light. Light is electromagnetic waves.

6.3.5. Electromagnetic field energy flow

As electromagnetic waves propagate through space and time, they carry energy with them. It is contained in mutually transforming electric and magnetic fields.

Volumetric electric field energy density

, (5.22)

where E is the electric field strength.

Volumetric magnetic field energy density

, (5.23)

where B is the magnetic field induction.

Consequently, the volumetric energy density of the electromagnetic field in the region of space where the electromagnetic wave is located at an arbitrary moment in time,

W= w e + w m =
. (5.24)

Or taking into account the fact that E = cB and
, we have

w =  o E 2 , (5.25)

or
. (5.26)

The energy transferred by an electromagnetic wave per unit time through a unit area is called the electromagnetic energy flux density. The electromagnetic energy flux density vector is called the Poynting vector.

Poynting vector direction coincides with the direction of propagation of the electromagnetic wave, i.e. with the direction of energy transfer. The speed of energy transfer is equal to the phase speed of this wave.

If an electromagnetic wave, when propagating, passes through a certain area S, perpendicular to the direction of its propagation, for example, along the X axis, then in a certain period of time dt the wave will travel a distance dx = cdt, where c is the speed of propagation of the wave.

Since the volumetric energy density of an electromagnetic wave

then the total energy dW of the electromagnetic wave contained in the volume

dW = wdV =  o E 2 cdtS. (5.27)

Consequently, the flux density of electromagnetic energy passing through the area S during the time dt

. (5.28)

Poynting vector coincides in direction with the speed of propagation of the electromagnetic wave, which is perpendicular to And , i.e.

. (5.29)

We use formula Stokes , according to which the circulation of the vector along a closed loop L is equal to the rotor flux of this vector through the surface resting on this contour. Then:

Let S — arbitrary time-invariant surface bounded by a contour L. Then the system of equations (1.2.7) will be rewritten as follows:

Since the contour of integration in the resulting integrals is arbitrary, the equality of the integrals to zero is possible only if the integrands are equal to zero. Then:

Equations (1.3.2) are Maxwell's equations.

In most of the course we will consider fields that change over time according to the harmonic law:

For which a complex recording form is adopted:

Where — complex amplitude. In the complex form of recording harmonic fields, the time derivative is replaced by multiplication by .

Then Maxwell's equations (1.3.2) for fields varying according to a harmonic law take the form:

Let's find a solution to Muswell's equations for the simplest case of propagation of an electromagnetic wave in a vacuum.

In a vacuum. Therefore, for vacuum, Maxwell’s equations (1.3.4) take the form:

Let us exclude From (1.3.5). To do this we apply the operation Rot To both sides of the first equation: . Now let's substitute the value from the second equation. As a result we get:

We use the well-known vector algebra relation

Let us remember that according to With Gauss-Ostrogradsky theorem

And let’s take into account that in a vacuum there are no free charges (i.e. ). Let's substitute (1.3.8) and (1.3.7) into (1.3.6). As a result we get:

The resulting equation is called Wave equation . In a similar way, one can obtain the wave equation for the magnetic field vector.

The most obvious solution to the wave equation is a spherical wave propagating around a point emitter. To obtain a solution for a spherical wave, it is necessary to represent the Laplace operator in equation (1.3.9) in a spherical coordinate system, which will lead to rather cumbersome mathematical expressions. In order to simplify mathematical procedures, we will consider solving the wave equation for a plane wave, which is a function of one coordinate.

Fig.1.3.1. shows a diagram of the location of the field lines of a spherical electromagnetic wave. The figure illustrates the fact that at large distances from the emitter, the electromagnetic field can be considered as a plane wave propagating along a direction perpendicular to the plane of constant phase, and the characteristics of the wave depend only on one coordinate along the direction of propagation. Despite the fact that in the general case the wave has spherical symmetry, in a limited region, indicated by a square, we can talk about a plane wave, the characteristics of which depend on only one coordinate.

Let us take into account that the one-dimensional Laplace operator has the following form:

And we get a one-dimensional wave equation for a plane wave:

Fig.1.3.1. Scheme of force lines of electric and magnetic fields of a spherical electromagnetic wave.

Any differential equation acquires physical meaning if boundary conditions for its solution are specified. The solution to equation (1.3.11) is obtained in the form of two waves propagating along the positive and negative directions of the z axis. Let us accept as boundary conditions the statement that in the medium under consideration a plane wave can propagate only in one direction. So, we have a solution to equation (1.3.11) for a plane wave propagating along the positive direction of the z axis:

Wave phase:

Where K— wave number (in general, a wave vector).

The fixed orientation of the field strength vector along a given coordinate axis is called Wave polarization . Relationship (1.3.12) specifies the polarization of the electric field strength along the axis X.

In Fig. 1.3.2. The position of the constant phase plane is shown for two moments in time.

Fig.1.3.2. Motion of a plane of constant phase.

For a constant phase plane ( φ = const), which moves along the z axis, its time derivative is zero:

In accordance with (1.1.26) we obtain:

Where is the speed of movement of the surface of a constant phase or Phase speed.

Substituting (1.3.12) into (1.3.11) we get

And, having reduced , we get Dispersion equation for a plane wave in free space:

Or (1.3.16)

Different signs in the expression for K correspond to waves propagating along the axis Z in different directions. In accordance with (1.3.14):

In free space, where C— speed of light.

Thus, from Maxwell’s equations it follows that the speed of light in free space is determined by the dielectric and magnetic permeability of vacuum:

Dielectric and magnetic permeability of vacuum are characteristics of space associated with static fields. The first of them characterizes only the dielectric properties of the medium. And the second is only magnetic properties. The result of solving Maskwell's equations, presented by formula (1.3.18), links together electrostatics, magnetostatics and the dynamic process of light propagation.

Indeed, the dielectric constant can be obtained experimentally by measuring the force of interaction between two known charges Q1 And Q2 located at a distance R from each other:

(Coulomb's law).

Magnetic permeability can be obtained by measuring the force of interaction between two conductors of length and current and, accordingly, located at a distance R from each other:

(Biot-Savart-Laplace law)

Thus, from a static experiment one can obtain a numerical value .

Consequently, Maxwell's equations make it possible to express the speed of light in terms of characteristics obtained using static measurements.

Maxwell's equations relate the electric field, magnetic field, and electromagnetic waves (light) together. The creation of the concept of the electromagnetic field and the formulation of the equations that describe it served as one of the most important starting points for physics of the 20th century.

(notes in italics)

1. Bias current

2. Maxwell's system of equations

3. EM waves and their characteristics

4. Obtaining EM waves - Hertz's experiments

5. Application of EM waves

1. In real life, there are no separate electric and magnetic fields; there is a single electromagnetic field.

The theory of the electromagnetic field, the beginning of which was laid by Faraday, was mathematically completed by Maxwell. An important idea put forward by Maxwell was the idea of symmetry in the interdependence of electric and magnetic fields. Namely, since a time-varying magnetic field (dB/dt) produces an electric field, we would expect a time-varying electric field (dE/dt) to produce a magnetic field.

According to the theorem on the circulation of the vector H

Let us apply this theorem to the case when a pre-charged flat capacitor is discharged through some external resistance (Fig. a).

For contour G, let's take a curve enclosing the wire. You can stretch different surfaces onto the contour Г, for example S and S. Both surfaces have “equal rights”, however, current I flows through the surface S, and through the surface S" no current. Surface S" “penetrates” only the electric field. By Gauss's theorem, the flow of vector D through a closed surface

D dS = q

According to the definition of current density, we have

Let's add the left and right sides of the equations, we get

From the equation it is clear that In addition to the conduction current density j, there is one more term dD/dt, the dimension of which is equal to the dimension of the current density.

Maxwell called this term density bias current:

J cm = dD/dt.

The sum of the conduction current and displacement current is called full current.

Total current lines are continuous, unlike conduction current lines. Conduction currents, if they are not closed, are closed by displacement currents.

It should be kept in mind that displacement current is equivalent to conduction current only in its ability to produce a magnetic field.

Displacement currents exist only where the electric field changes over time. In essence, he himself is an alternating electric field.

Maxwell's discovery of the displacement current is a purely theoretical discovery, and of paramount importance.

2. With the introduction of displacement current, the macroscopic theory of the electromagnetic field was completed. Opening bias current ( dD/dt) allowed Maxwell to create a unified theory of electrical and magnetic phenomena. Maxwell's theory not only explained all the disparate phenomena of electricity and magnetism, but also predicted a number of new phenomena, the existence of which was later confirmed.

Maxwell's electromagnetic theory is based on four fundamental equations of electrodynamics, called Maxwell's equations.

These equations express in a condensed form the entirety of our knowledge about the electromagnetic field.

1. The circulation of vector E along any closed contour is equal with a minus sign to the time derivative of the magnetic flux through any surface limited by this contour. In this case, E is understood not only as a vortex electric field, but also as an electrostatic one.

2. The flow of vector B through an arbitrary closed surface is always zero.

3. The circulation of vector H along any closed circuit is equal to the total current (conduction current and displacement current) through an arbitrary surface limited by this circuit.

4. The flux of vector D through any closed surface is equal to the algebraic sum of the external charges covered by this surface.

From Maxwell's equations for the circulation of vectors E and H it follows that the electric and magnetic fields cannot be considered as independent: a change in time of one of these fields leads to the appearance of the other. Therefore, only the totality of these fields, which describes a single electromagnetic field, makes sense.

These equations indicate that an electric field can arise for two reasons. Firstly, its source is electrical charges, both external and bound. Secondly, field E is always formed when the magnetic field changes over time.

These same equations indicate that the magnetic field B can be excited either by moving electric charges (electric currents), or by alternating electric fields, or by both at the same time. There are no sources of magnetic field similar to electric charges in nature, this follows from the second equation.

The significance of Maxwell’s equations is not only that they express the basic laws of the electromagnetic field, but also that by solving (integrating) them the fields E and B themselves can be found.

Maxwell's equations are more general; they are also valid in cases where there are fracture surface - surfaces on which the properties of the medium or fields change abruptly.

Maxwell's fundamental equations do not yet constitute a complete system of electromagnetic field equations. These equations are not enough to find fields from given distributions of charges and currents. They must be supplemented with relations, these relations are called material equations.

The material equations are most simple in the case of fairly weak electromagnetic fields that change relatively slowly in space and time. In this case, for isotropic media, the material equations have the following form:

=εε 0

=μμ 0

=γ( + st)

Maxwell's equations have a number of properties.

1 properties – linearity.

Maxwell's equations are linear because they contain only the first derivatives of the fields E and B with respect to time and spatial coordinates and the first degrees of the density of electric charges and currents.

The linearity property of Maxwell's equations is directly related to the principle of superposition: if any two fields satisfy Maxwell's equations, then this also applies to the sum of these fields.

2nd property – continuity.

Maxwell's equations contain a continuity equation expressing the law of conservation of electric charge.

3 property – invariance.

Maxwell's equations are satisfied in all inertial frames of reference. They are relativistically invariant. This is a consequence of the principle of relativity, according to which all inertial frames of reference are physically equivalent to each other. The fact of invariance of Maxwell's equations is confirmed by numerous experimental data.

Maxwell's equations are correct relativistic equations, unlike, for example, Newton's equations of mechanics.

4th property – symmetry.

Maxwell's equations are not symmetrical with respect to electric and magnetic fields. This is due to the fact that in nature there are electric charges, but no magnetic charges.

In a neutral, homogeneous, non-conducting medium, Maxwell's equations take on a symmetrical form.

From Maxwell's equations it follows that there is a fundamentally new physical phenomenon: the electromagnetic field can exist independently - without electric charges and currents. In this case, the change in its state necessarily has a wave character. Fields of this kind are called electromagnetic waves. In a vacuum they always propagate at a speed equal to the speed c.

It also turned out that the displacement current (dD/dt) plays a primary role in this phenomenon. It is its presence, along with the dB/dt value, that means the possibility of the appearance of electromagnetic waves. Any change in time of the magnetic field excites an electric field, and a change in the electric field, in turn, excites a magnetic field.

Due to continuous mutual transformation or interaction, they must be preserved - the electromagnetic disturbance will spread in space.

Maxwell's theory not only predicted the possibility of the existence of electromagnetic waves, but also made it possible to establish all their basic properties.

3. The existence of electromagnetic waves was theoretically predicted by the great English physicist J. Maxwell in 1864.

Maxwell's hypothesis was only a theoretical assumption that did not have experimental confirmation, but on its basis Maxwell managed to write down a consistent system of equations describing the mutual transformations of electric and magnetic fields, that is, a system of equations electromagnetic field(Maxwell's equations). A number of important conclusions follow from Maxwell's theory, one of them was the conclusion about the existence of electromagnetic waves.

Electromagnetic waves transverse– the vectors are perpendicular to each other and lie in a plane perpendicular to the direction of wave propagation(rice.).

Electromagnetic waves propagate in matter at a finite speed

The speed c of propagation of electromagnetic waves in a vacuum is one of the fundamental physical constants.

4. Maxwell argued that electromagnetic waves have the properties of reflection, refraction, diffraction, etc. But any theory becomes proven only after it is confirmed in practice. But at that time, neither Maxwell himself nor anyone else knew how to experimentally obtain electromagnetic waves. This happened only after 1888, When Hertz experimentally discovered electromagnetic waves.

As a result of experiments, Hertz created a source of electromagnetic waves, which he called a “vibrator”. The vibrator consisted of two conducting spheres(in a number of experiments cylinders) with a diameter of 10-30 cm, fixed at the ends of a wire rod cut in the middle. The ends of the rod halves at the cut site ended in small polished balls, forming a spark gap of several millimeters.

The spheres were connected to the secondary winding of the Ruhmkorff coil, which was a source of high voltage.

From Maxwell's theory it is known

1) only an accelerated moving charge can emit an electromagnetic wave,

2) that the energy of an electromagnetic wave is proportional to the fourth power of its frequency.

It is clear that charges move at an accelerated rate in an oscillatory circuit, so the easiest way is to use them to emit electromagnetic waves. But it is necessary to make sure that the frequency of charge oscillations becomes as high as possible. From Thomson's formula for the cyclic frequency of oscillations in a circuit it follows that to increase the frequency it is necessary to reduce the capacitance and inductance of the circuit.

To reduce the capacitance C it is necessary to increase the distance between the plates(move them apart, make the outline open) and reduce the area of the plates. The smallest capacity that can be obtained is just a wire.

To reduce the inductance L it is necessary to reduce the number of turns. As a result of these transformations we get just a piece of wire or open oscillatory circuit OCC.

The essence of the phenomena occurring in the vibrator is as follows. The Ruhmkorff inductor creates a very high voltage, on the order of tens of kilovolts, at the ends of its secondary winding, which charges the spheres with charges of opposite signs. At a certain moment, an electric spark appears in the spark gap of the vibrator, making the resistance of its air gap so small that high-frequency damped oscillations arise in the vibrator, lasting as long as the spark exists. Since the vibrator is an open oscillatory circuit, electromagnetic waves are emitted.

After a huge series of labor-intensive and extremely cleverly staged experiments using the simplest, so to speak, available means, the experimenter achieved his goal. It was possible to measure the wavelengths and calculate the speed of their propagation. have been proven

· presence of reflection,

· refraction,

· diffraction,

interference and polarization of waves.
electromagnetic wave speed measured

5. Electromagnetic waves were first used seven years after Hertz's experiments. On May 7, 1895, A. S. Popov (1859-1906), a teacher of physics for officer mine classes, at a meeting of the Russian Physico-Chemical Society demonstrated the world's first radio receiver, which opened up the possibility of the practical use of electromagnetic waves for wireless communication, which transformed the life of mankind. The first radiogram transmitted in the world contained only two words: “Heinrich Hertz.” The invention of radio by Popov played a huge role in the spread and development of Maxwell's theory.

Electromagnetic waves of the centimeter and millimeter ranges, encountering obstacles in their path, are reflected from them. This phenomenon is the basis of radar - detecting objects (for example, aircraft, ships, etc.) over long distances and accurately determining their position. In addition, radar techniques are used to observe the passage and formation of clouds, the movement of meteorites in the upper atmosphere, etc.

Electromagnetic waves are characterized by the phenomenon of diffraction - the bending of waves around various obstacles. It is thanks to the diffraction of radio waves that stable radio communication is possible between remote points separated by the convexity of the Earth. Long waves (hundreds and thousands of meters) are used in phototelegraphy, short waves (several meters or less) are used in television to transmit images over short distances (a little more than line-of-sight limits). Electromagnetic waves are also used in radio geodesy for very precise determination of distances using radio signals, in radio astronomy for studying the radio emission of celestial bodies, etc. It is almost impossible to give a complete description of the use of electromagnetic waves, since there are no areas of science and technology where they are not were used.

To carry out radio and television communications, electromagnetic waves with a frequency from several hundred thousand hertz to hundreds of megahertz are used.

When transmitting speech, music and other sound signals via radio, various types of modulation of high-frequency (carrier) oscillations are used. The essence of modulation is that high-frequency oscillations generated by the generator change according to the law of low frequency. This is one of the principles of radio transmission. Another principle is the reverse process - detection. When receiving radio signals, it is necessary to filter out low-frequency sound vibrations from the modulated signal received by the antenna of the receiver.
With the help of radio waves, not only sound signals are transmitted over a distance, but also images of objects.

Related information.

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 ):

The quantities and are the electric and magnetic constants, respectively, which are related to the speed of light in vacuum by the relation

Constants characterize the electrical and magnetic properties of the medium, 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.

From Maxwell's equations we obtain 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 according to certain rules on 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:

Let's express rot B through an electric field using Maxwell's equation:

and use this expression on the right side of (2.93). As a result, we arrive at the equation:

Considering the connection

and entering refractive index environment

Let's write the equation for the electric field strength vector in the form:

Comparing with (2.69), we are convinced that we have obtained the wave equation, where v- phase speed of light in the medium:

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:

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:

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:

Substituting here expressions (2.99), which depend only on the coordinate x, we find:

Differentiating plane waves with respect to time gives:

Then from Maxwell’s equations it follows:

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:

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:

This leads to the usual relationship between wave vector, frequency and speed:

as well as the connection between the amplitudes of field oscillations:

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 wavelength ranges of different types overlap each other. Therefore, waves of such lengths can be obtained in various 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.

Additional Information

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:

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:

Let's substitute these expressions into the expression for phase, to get the phase waves in a moving frame of reference:

This expression can be written as

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:

For an 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:

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:

If , then the observer approaches the source and the radiation frequency for it increases:

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:

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 speed of the wave

in a moving reference frame. From the Lorentz transformations for the wave vector we have:

Let's substitute the ratio here:

We get:

From here we find the wave speed in the moving frame of reference:

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: