Maxwell's equations and 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 electrical oscillations. magnetic 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 proportional to the rate of change magnetic flux through a surface stretched over a contour (this is Faraday’s law of electromagnetic induction):

(1)

Physical meaning of this equation - a changing magnetic field generates 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 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:

In electrodynamics, these are like Newton’s laws in classical mechanics or Einstein’s postulates in the theory of relativity. Fundamental equations, the essence of which we will understand today so as not to fall into a stupor at their mere mention.

Useful and interesting information on other topics - in our telegram.

Maxwell's equations are a system of equations in differential or integral form that describes any electromagnetic fields, the relationship between currents and electric charges in any media.

They were reluctantly accepted and critically perceived by Maxwell's contemporaries. This is because these equations were not similar to anything from known to people previously.

Nevertheless, to this day there is no doubt about the correctness of Maxwell’s equations; they “work” not only in the macroworld we are familiar with, but also in the field of quantum mechanics.

Maxwell's equations made a real revolution in people's perception of the scientific picture of the world. Thus, they anticipated the discovery of radio waves and showed that light is electromagnetic in nature.

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Let us write and explain all 4 equations in order. Let us immediately clarify that we will write them in the SI system.

The modern form of Maxwell's first equation is:

Here we need to explain what divergence is. Divergence is a differential operator that determines the flow of a field through a certain surface. A comparison with a tap or a pipe would be appropriate. For example, the larger the diameter of the faucet spout and the pressure in the pipe, the greater the flow of water through the surface that the spout represents.

In Maxwell's first equation E is a vector electric field, and the Greek letter “ ro » – the total charge contained inside a closed surface.

So, the electric field flow E through any closed surface depends on the total charge inside that surface. This equation is Gauss' law (theorem).

Maxwell's third equation

Now we will skip the second equation, since Maxwell's third equation is also Gauss's law, only not for the electric field, but for the magnetic one.

It looks like:

What does it mean? The magnetic field flux through a closed surface is zero. If electric charges (positive and negative) can well exist separately, generating an electric field around themselves, then magnetic charges simply do not exist in nature.

Maxwell's second equation is nothing more than Faraday's law. Its appearance:

The rotor of the electric field (the integral through a closed surface) is equal to the rate of change of the magnetic flux penetrating this surface. To better understand, let's take the water in the bathroom that is drained through a hole. A funnel forms around the hole. Rotor is the sum (integral) of the velocity vectors of water particles that rotate around the hole.

As you remember, based on Faraday's law Electric motors operate: a rotating magnet generates a current in a coil.

The fourth is the most important of all Maxwell's equations. It was there that the scientist introduced the concept bias current.

This equation is also called the theorem on the circulation of the magnetic induction vector. It tells us that electric current and changes in the electric field generate a vortex magnetic field.

Let us now present the entire system of equations and briefly outline the essence of each of them:

First equation: electric charge generates an electric field

Second equation: a changing magnetic field generates a vortex electric field

Third equation: there are no magnetic charges

Fourth equation: electric current and changes in electrical induction generate a vortex magnetic field

Solving Maxwell's equations for free electromagnetic wave, we get the following picture of its distribution in space:

We hope this article will help systematize knowledge about Maxwell's equations. And if you need to solve a problem in electrodynamics using these equations, you can safely turn to the student service for help. Detailed explanation any assignment and an excellent grade are guaranteed.

By group differential equations. The differential equations that each of the field vectors must satisfy separately can be obtained by eliminating the remaining vectors. For the field region that does not contain free charges and currents ($\overrightarrow(j)=0,\ \rho =0$), the equations for the vectors $\overrightarrow(B)$ and $\overrightarrow(E)$ have the form:

Equations (1) and (2) are ordinary equations of wave motion, which indicate that light waves propagate in a medium with a speed ($v$) equal to:

Note 1

It should be noted that the concept of the speed of an electromagnetic wave has a certain meaning only in connection with waves simple type, for example flat. The speed $v$ is not the speed of wave propagation in the case of an arbitrary solution to equations (1) and (2), since these equations admit solutions in the form of standing waves.

In any wave theory of light, a harmonic wave in space and time is considered an elementary process. If the frequency of this wave lies in the interval $4\cdot (10)^(-14)\frac(1)(c)\le \nu \le 7.5\cdot (10)^(-14)\frac(1) (c)$, such a wave causes a physiological sensation of a certain color in a person.

For transparent substances, the dielectric constant $\varepsilon $ is usually greater than unity, the magnetic permeability of the medium $\mu $ is almost equal to unity, it turns out that, in accordance with equation (3), the speed $v$ is less than the speed of light in vacuum. What was experimentally shown for the first time for the case of light propagation in water by scientists Foucault And Fizeau.

Usually it is not the velocity value itself that is determined ($v$), but the ratio $\frac(v)(c)$, for which they use law of refraction . In accordance with this law, when a plane electromagnetic wave is incident on a flat boundary that separates two homogeneous media, the ratio of the sine of the angle $(\theta )_1$ of incidence to the sine of the angle of refraction $(\theta )_2$ (Fig. 1) is constant and equal to ratio of wave propagation velocities in two media ($v_1\ and (\v)_2$):

Meaning permanent relationship expressions (4) are usually denoted as $n_(12)$. They say that $n_(12)$ is the relative refractive index of the second substance in relation to the first, which the wave front (wave) experiences when passing from the first medium to the second.

Picture 1.

Definition 1

Absolute refractive index(simply the refractive index) of a $n$ medium is the refractive index of a substance relative to vacuum:

A substance with a higher refractive index is optically denser. The relative refractive index of two substances ($n_(12)$) is related to their in absolute terms($n_1,n_2$) like:

Maxwell's formula

Definition 2

Maxwell found that the refractive index of a medium depends on its dielectric and magnetic properties. If we substitute the expression for the speed of light propagation from equation (3) into formula (5), we get:

\ \

Expression (7) is called Maxwell's formula. For most non-magnetic transparent substances that are considered in optics, the magnetic permeability of the substance can be approximately considered equal to one, therefore equality (7) is often used in the form:

It is often assumed that $\varepsilon$ is constant. However, we are well aware of Newton's experiments with a prism on the decomposition of light; as a result of these experiments, it becomes obvious that the refractive index depends on the frequency of light. Consequently, if we assume that Maxwell’s formula is valid, then we should recognize that the dielectric constant of a substance depends on the field frequency. The connection between $\varepsilon $ and the field frequency can only be explained if we take into account the atomic structure of the substance.

However, it must be said that Maxwell's formula with a constant dielectric constant of a substance can in some cases be used as a good approximation. An example is gases with a simple chemical structure, in which there is no significant dispersion of light, which means that the optical properties are weakly dependent on color. Formula (8) also works well for liquid hydrocarbons. On the other hand, the majority solids, for example, glass and most liquids exhibit a strong deviation from formula (8), if we consider $\varepsilon$ constant.

Example 1

Exercise: What is the concentration of free electrons in the ionosphere if it is known that for radio waves with a frequency $\nu$ its refractive index is equal to $n$.

Solution:

Let's take Maxwell's formula as a basis for solving the problem:

\[\varepsilon =1+\varkappa =1+\frac(P)((\varepsilon )_0E)\left(1.2\right),\]

where $\varkappa$ is the dielectric susceptibility, P is the instantaneous polarization value. From (1.1) and (1.2) it follows that:

If the concentration of atoms in the ionosphere is equal to $n_0,$ then the instantaneous value of polarization is equal to:

From expressions (1.3) and (1.4) we have:

where $\omega $ is the cyclic frequency. The equation of forced oscillations of an electron without taking into account the resistance force can be written as:

\[\ddot(x)+((\omega )_0)^2x=\frac(q_eE_0)(m_e)cos\omega t\left(1.7\right),\]

where $m_e$ is the mass of the electron, $q_e$ is the charge of the electron. The solution to equation (1.7) is the expression:

\ \

We know the frequency of radio waves, therefore we can find the cyclic frequency:

\[\omega =2\pi \nu \left(1.10\right).\]

Let's substitute in (1.5) right side expression (1.9) instead of $x_(max)$ and use (1.10), we get:

Answer:$n_0=\frac(E_0m_e4\pi ^2\nu ^2)((q_e)^2)\left(1-n^2\right).$

Example 2

Exercise: Explain why Maxwell's formula contradicts some experimental data.

Solution:

From Maxwell's classical electromagnetic theory it follows that the refractive index of a medium can be expressed as:

where in the optical region of the spectrum for most substances we can assume that $\mu \approx 1$. It turns out that the refractive index for a substance must be a constant value, since $\varepsilon $ - the dielectric constant of the medium is constant. Whereas experiment shows that the refractive index depends on frequency. The difficulties that Maxwell's theory encountered in this matter are eliminated by Lorentz's electronic theory. Lorentz considered the dispersion of light as a 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 light wave. Using his hypothesis, Lorentz obtained a formula relating the refractive index to the frequency of an electromagnetic wave (see example 1).

Answer: The problem with Maxwell's theory is that it is macroscopic and does not consider the structure of matter.

In microwave technology, the interest is mainly in fields that vary with time according to a harmonic law (i.e., they are sinusoidal in nature).

Using the complex method, we write the vectors of the electric and magnetic fields:

,
, (33)

Where – angular frequency
.

Let's substitute these expressions into I and II – Maxwell's equations

,
.

After differentiation we have:

, (34)

. (35)

Equation (34) can be transformed to the form:

,

Where
– complex relative dielectric constant taking into account losses in the medium.

The ratio of the imaginary part of the complex relative dielectric constant to the real part represents the dielectric loss tangent
. Thus, Maxwell's equations for harmonic vibrations in the absence of free charges
have the form:

,(36)

, (37)

, (38)

. (39)

In this form, Maxwell's equations are inconvenient and must be transformed.

Maxwell's equations are easily reduced to wave equations, which include only one of the field vectors. Defining
from (37) and substituting it into (36), we obtain:

Let's expand the left side using formula III:

Let us introduce the notation
, then taking into account
, we get:

. (40)

The same equation can be obtained for

. (41)

Equations (40) – (41) are called Helmholtz equations. They describe the propagation of waves in space and are proof that changes in electric and magnetic fields over time lead to the propagation of electromagnetic waves in space.

These equations are valid for any coordinate system. When using a rectangular coordinate system we will have:

, (42)

, (43)

Where
– unit vectors

If we substitute relations (42) and (43) into equations (40) and (41), then the latter break down into six independent equations:

,
,

, (44)
, (45)

,
,

Where
.

In the general case, in a rectangular coordinate system, to find the field components, it is necessary to solve one second-order linear differential equation

,

Where – one of the components of the field, i.e.
. The general solution to this equation is

, (46)

Where
– field distribution function in the plane of the wave front, independent of .

Energy relationships in the electromagnetic field. Umov-Poynting theorem

One of the most important characteristics of the electromagnetic field is its energy. For the first time, the question of the energy of the electromagnetic field was considered by Maxwell, who showed that the total energy of the field contained inside a volume , consists of the energy of the electric field:

, (47)

and magnetic field energy:

. (48)

Thus, the total energy of the electromagnetic field is equal to:

. (49)

In 1874 prof. N.A. Umov introduced the concept of energy flow, and in 1880. this concept was applied by Poynting to the study of electromagnetic waves. The radiation process in electrodynamics is usually characterized by determining the Umov-Poynting vector at each point in space.

Physically correct results, consistent with both the law of conservation of energy and Maxwell’s equations, are obtained if we express the Umov-Poynting vector in terms of instantaneous values
And
in the following way:

.

Let's take Maxwell's first and second equations and multiply the first by , and the second on
and add:

,

Where .

Thus, equation (50) can be written as

,

integrating over volume and changing signs, we have:

Let us move from the integral over the volume to the integral over the surface

,

or taking into account
we get:

, That
,
,

. (51)

The resulting equation expresses the law of conservation of energy in an electromagnetic field (Umov-Poynting theorem). The left side of the equation represents the rate of change over time of the total energy reserve electromagnetic field in the considered volume
. The first term on the right side is the amount of heat , released in the conducting parts of the volume per unit of time. The second term represents the flow of the Umov-Poynting vector through the surface bounding the volume .Vector
is the energy flux density of the electromagnetic field. Because
, then the direction of the vector
can be determined by the vector product rule /gimlet rule/ (Fig. 9). In system SI vector
has dimension
.

Figure 9 – Towards the definition of the Umov-Poynting vector

The basic equations of classical electrodynamics (Maxwell's system of equations) are rightfully generally accepted equations and are widely used in physics, radiophysics and electronics. However, these equations were not obtained from general physical laws, which did not allow them to be considered absolutely accurate and allowed various kinds of manipulation with them. However, these equations are exact and are derived from general principles physics and fundamentals of vector algebra.

1. Derivation of Faraday’s law of electromagnetic induction

Faraday's law of electromagnetic induction can be obtained from the equation for electromagnetic forces acting on a point electric charge:

This situation occurs in Explorer with electric shock high frequency, when the force acting on the electron from the primary electric field changes so quickly that it is in antiphase with the inertial force of the electrons.

Let us reduce the charge in equality (2) and apply the “rotor” operation to both sides of this equality:

.

(3)

Let, for example, the axis z coincides with the direction of the axial vector B , then the radius vector will look like: r =x i+y j , Where i And j – unit vectors in the directions of the coordinate axes x And y, respectively. Radial vector r has no third component along the axis z, therefore the second term in (3) is equal to –2(∂ B /∂t). The first term in equation (3) is equal to ∂ B /∂t. As a result, after transforming the right side of the last equality, we get:

.

(4)

That is, from the electromagnetic force equation (1) in the case when the force acting on the electron from the magnetic field is completely balanced by the force from the electric field, Faraday’s law of electromagnetic induction (4) follows, one of the basic equations of electrodynamics.

Equations (2) – (4) do not depend on whether an electron is present or absent at a given point in space. As a result of this independence of the electric and magnetic fields from the electric charge, equation (4) reflects the spatiotemporal properties of the changing fields themselves, represented as a single electromagnetic field. Moreover, Faraday’s law (4) not only represents the law of electromagnetic induction, but is also the basic law of mutual transformation of electric and magnetic fields, an integral property of the electromagnetic field.

2. Derivation of Maxwell's equation

Before proceeding to the derivation of Maxwell's equation, it is necessary to supplement vector algebra with another vector operator.

2.1. Definition of a vector operator that performs the inverse action of the vector transformation of the differential vector operator “rotor”

The differential vector operator “rotor” performs the operation of transforming vectors in space and the operation of differentiation, that is, it is a complex operator that performs two types of actions at once. This follows directly from its definition:

,

Where A – vector, i , j , k – unit vectors in the direction of the axes of the rectangular (Cartesian) coordinate system x, y And z, respectively. In this case, the operator inverse to the “rotor” operator is not defined in vector analysis, although each of the transformations it performs is, in principle, invertible.

Geometric vector spatial transformation illustration A to vector rot( a) , carried out by the “rotor” operator, is shown in Fig. 1.

Rice. 1. Geometric representation of a vector A and the vector field formed by the “rotor” operator.

2.2. Definition 1. If two interrelated vector fields represented by vectors A And b , have derivatives with respect to spatial variables x, y, z(as rot a And rot b ) and derivatives with respect to time, ¶ A /¶ t And ¶ b /¶ t, and the derivative of the vector A is orthogonal in time to the derivatives with respect to the spatial variables of the vector b , and vice versa, the time derivative of the vector b orthogonal to the derivatives with respect to the spatial variables of the vector A , then there is a vector operator that carries out a spatial transformation of the vector field without affecting the differentiation operation, which we will conventionally call the operator “ rerot", (oppositely twisted or "reversible rotor") such that:

And ; (5)

And . (5*)

2.3. Properties of the vector operator “reversible” rotor"

2.3.1. The vector operator "reversible rotor" acts only on derivatives of a vector.

2.3.2. The vector operator "reversible rotor" is located before the derivative of the vector on which it acts.

2.3.3. Constants and numerical coefficients for vector derivatives can be moved outside the scope of vector operators:

Where c- constant.

2.3.4. The vector operator “reversible rotor” acts on each of the terms of the equation containing the sum of vector derivatives:

Where c And d- constants.

2.3.5. The result of the action of the vector operator “reversible rotor” on zero is zero:

In this case, the result of the action of the vector operator “reversible rotor” on other constants, including the vector, according to paragraph 2.3.1, is not defined.

2.4. An example of using the “reversible rotor” operator

Let us apply the “reversible rotor” operator to an equation containing interconnected vectors a And b :

If we now apply the “reversible rotor” operator again to the newly formed equality (**), we obtain:

or

, or finally:

.

((*))

Successive double (or any even) application of the reverse rotor operator results in the original equality. By this, the vector operator “reversible rotor” not only carries out the mutual transformation of differential equations of interconnected vector fields, but also establishes the equivalence of these equations.

Geometrically it looks like this. The “rotor” operator differentiates and, as it were, twists a rectilinear vector field, making it vortex and orthogonal to the original vector field. The vector operator “reversible rotor” performs a vector transformation, which, as it were, unwinds the vortex field twisted by the “rotor” operator, turning it into a changing non-vortex field, represented by the derivative of the vector with respect to time. Since integration is not performed, the derivative of the vector with respect to time corresponds to the change in the magnitude of the vector. As a result, we have a change in vector, the magnitude of which changes in a single direction, orthogonal to the spatial variables of the “rotor” operator. Conversely, the “reversing rotor” vector operator spins the non-vortex changing vector field represented by the vector’s time derivative, turning it into an eddy spatial vector field orthogonal to the original time derivative of the vector. Since the direction of “torsion” of the “reversible rotor” operator is opposite to the direction of rotation carried out by the “rotor” operator, the sign of the newly formed vortex field is chosen to be opposite (negative). That is, the vector operator “reversible rotor” performs the inverse action of the spatial transformation of the operator “rotor” on the entire “space” of derivative vector fields. At the same time, the vector operator “reversible rotor” does not itself differentiate the vector on whose derivative it acts. This results in an identical reversible vector transformation.

If we introduce into vector analysis an integral vector operator that restores not the derivative of the vector, but the vector itself from the rotor of the vector (let’s conventionally call such an operator an inverse rotor, or “ rot-1 "), then such an operator, along with the inverse vector transformation, must simultaneously perform the integration operation.

However, due to the ambiguity mathematical operation integration, completely inverse to the “rotor” operator rot-1 does not perform a unique inverse vector transformation.

2.5. Application of the vector operator "reversible" rotor" to physical fields

When applying the “reversible rotor” vector operator to physical vector fields, it is necessary to take into account the change in the dimension of the right and left sides of the equation due to the permutation of variables x, y, z And t when converting. Let us denote the dimension of the coordinates – meter ( L), and time is second ( T).

Definition 2. For physical vector fields, the vector operator “reversible rotor” is defined as follows:

And ;

(6)

And . (6*)

Denoting dimensional relationship L/T, as a constant v, having the dimension of speed, [m/s], equations (6.4) and (6.4*) can be represented as:

And ;

(7)

And .

(7*)

2.6. Application of the “reversible rotor” operator to physical fields

Let us apply the vector operator “reversible rotor”, defined by equations (7), (7*), to equation (4), connecting real physical fields E And B in electrodynamics:

;

, which transforms to the form:

(8)

>.

Electrodynamic constant " v» does not depend on the magnitude of the fields or on the rate of their change and, as follows from the wave equation, corresponds to the speed of propagation of the wave of electromagnetic interaction, c" 2.99792458H 10 8 m/s, which is also called the speed of light in vacuum.

That is, with the help of the “reversible rotor” vector transformation, from equation (4), which is Faraday’s law of electromagnetic induction, one of the basic equations of electrodynamics naturally follows - Maxwell’s equation (8), which does not follow either from experiment or from known physical laws . Equations (4) and (8) are interrelated, transformable into each other using a vector transformation, which corresponds to their physical equivalence. Therefore, the validity of one of these equations, established in the form of a physical law (in in this case- this is Faraday's law of electromagnetic induction (4)) is a sufficient condition for asserting the validity of the second equation (Maxwell's equation (8)) as an equivalent physical law.

2.7. Transformation of vector fields

If we proceed from the definition of the “rotor” operator, then the action of the “reverse rotor” vector operator, it would seem, can be represented in the form shown in Fig. 2, where some identity of the vector fields is assumed before and after the vector transformation by the differential vector operator “rotor”.

Let's check this assumption. Let us apply the “reversible rotor” operator to the equation:

, from which it follows:

The resulting equality changes the direction of the vectors in the original definition of the differential vector operator “rotor,” which is unacceptable.

That's why .

The application of the vector operator “reversible rotor” to derivatives of the same vector field shows the fundamental difference between the vector field before application and the vector field after application of the “rotor” operator. This means the need to represent the vector field A and vector field rot( A) as transformable into each other, but different vector fields.

The original vector field represented by the vector A , we will consider the primary (cause), and the field formed by the vector transformation of the “rotor” operator will be considered a secondary field (a consequence of the action of the “rotor” operator) and denote it as a field of vectors b .

Rice. 2. The result of identifying vector fields before and after the “rotor” vector transformation. The direction of the fields does not correspond to the original definition of the rotor operator shown in Fig. 1, “right screw” turns into “left screw”.

Then inverse conversion vector fields, which does not affect the operation of differentiation, in the notation introduced in this way will have the form shown in Fig. 3.

Rice. 3. Definition of a vector transformation inverse to the “rotor” operation, which does not affect the differentiation operation. The division of vector fields is carried out on the basis of cause-and-effect relationships. The original field is represented by the vector A (cause), and the field generated by the “rotor” operation is represented by the vector b (consequence).

In electrodynamics, in some of the simplest cases, the transition to a rotating reference frame, within which rotation disappears, leads to the absence of forces from the magnetic field, and the force action can be represented only by the force from the electric field. But this does not in any way lead to the conclusion that there is no magnetic field or that it can always be replaced by an electric field. Special case vector field, taken in a separate isolated reference system, applies only to this selected system in which the movement of an electric charge is limited in degrees of freedom.

Since both rectilinear vector fields and rotating closed vector fields exist in space, and it is impossible to be in two reference systems at the same time, then in the general case, by choosing a coordinate system it is impossible to reduce one field to another. There is only one source of these fields - electrical charges. Electric charges create an electric field around themselves (omnidirectional vector field), and the movement of electric charges creates a magnetic field (closed circular vector field). In this case, naturally, the rectilinear motion of electric charges creates a circular magnetic field around them, and the circular motion of electric charges (as well as the rotation of electrically charged particles around their own axis) creates a magnetic field rectilinear in space, contained in a volume limited by the radius of rotation.

2.8. Speed ​​of propagation of electromagnetic interaction

The rate of transformation of vector fields into each other does not depend on either the magnitude of the fields or the rate of their change and, as follows from the wave equation, corresponds to the speed of propagation of a wave of electromagnetic interaction in free space (vacuum), c" 2.99792458Х 10 8 m/s, and this value is rightly called the electrodynamic constant.

Thus, the change in electric and magnetic fields carried out in three-dimensional space has the property of mutual transformation of vectors, and this property in electrodynamics is realized through Faraday’s law of electromagnetic induction. If we consider such a transformation to be direct, then the inverse transformation of vector fields is carried out using the equation obtained by Maxwell intuitively, and which can be obtained using the “reversible rotor” vector operator. The mutual transformation of electric and magnetic fields, which is carried out without sources of electric charge, is one of the special types of wave motion - a transverse electromagnetic wave, which transfers electromagnetic energy in free space with the absolute speed of field transformation. But at the same time, the source of energy of an electromagnetic wave is always accelerated moving electric charges.

3. Equations of sources of electromagnetic fields.

The remaining two of the four basic equations of Maxwell’s system of equations only establish the fact of the presence in nature of electric charges that create an electric field (Gauss’s theorem, which directly follows from Coulomb’s law):

and the fact that there are no magnetic charges in nature:

Literature

1. Sokol-Kutylovsky O.L. Gravitational and electromagnetic forces. Ekaterinburg, 2005.
2. Sokol-Kutylovsky O.L. Russian physics. Ekaterinburg, 2006.
3. Bronshtein I.N., Semendyaev K.A. Handbook of mathematics for engineers and students of technical colleges (edited by G. Groshe and V. Ziegler), M., “Nauka”, 1980.

Sokol-Kutylovsky O.L., Derivation of the basic equations of electrodynamics // “Academy of Trinitarianism”, M., El No. 77-6567, pub. 13648, 08/11/2006

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