In electrodynamics, it’s like Newton’s laws in classical mechanics or like 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.

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Maxwell's equations are a system of equations in differential or integral form, describing 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 scientific picture peace. 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 - This differential operator, which determines the flow of some 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 the vector electric field, and 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 represents 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? Magnetic field flux through a closed surface equal to zero. If electric charges (positive and negative) can well exist separately, generating an electric field around themselves, then magnetic charges simply does not exist in nature.

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

Electric field rotor (integral through a closed surface) equal to speed changes magnetic flux piercing 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 a free electromagnetic wave, we obtain the following picture of its propagation 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.

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 do 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 alternating electric field is always associated 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 propagation speed electromagnetic field according to 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 The strengths of the electric and magnetic fields 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 vibrations of the electric and magnetic vectors in an electromagnetic wave occur with the same phase.

A group of differential equations. Differential equations, which each of the field vectors must satisfy separately, can be obtained by excluding the remaining vectors. For a field area 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 mean that light waves propagate in the 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 arbitrary decision equations (1) and (2), since these equations admit solutions in the form of standing waves.

Anytime wave theory light is considered an elementary process harmonic wave in space and time. 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 falls on a plane 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 the ratio of the velocities of wave propagation in two media ($v_1\ and (\v)_2$) :

The value of the constant ratio of expression (4) is 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.

Figure 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 that has higher rate refraction is optically denser. Relative indicator refraction of two substances ($n_(12)$) is associated with 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 value. 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. Therefore, 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 atomic structure substances.

However, it must be said that Maxwell’s formula with a constant dielectric constant substances, in some cases can be used as a good approximation. An example would be gases with simple chemical structure, in which there is no significant dispersion of light, which means a weak dependence of optical properties 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 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 $n_0,$ 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 the classic electromagnetic theory Maxwell it follows that the refractive index of the 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. Difficulties that arose before Maxwell's theory in this issue, eliminates electron theory Lorenz. 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 light waves in an alternating electromagnetic field. 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.

Maxwell's equations and wave equation

Electromagnetic waves

During the propagation of a mechanical wave in elastic medium V oscillatory motion particles of the medium are involved. The reason for this process is the presence of interactions between molecules.

Besides elastic waves in nature there is a wave process of a different nature. It's about 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-ray radiation, γ – rays. A special place in this diversity occupies visible part 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 any 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, Maxwell's prediction material object– EMF – was discovered experimentally by Hertz.

Maxwell's equations are postulates of electrodynamics, formulated based on the analysis experienced 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 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 with terminal speed, that is, an electromagnetic wave arises. In order to prove the existence of wave process, 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 connection between the characteristics of fields in a homogeneous isotropic environment: 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 system of equations includes four basic equations

, (3.2)

, (3.3)

. (3.4)

This system is complemented by three material equations, defining the connection between physical quantities, included in Maxwell's equations:

(3.5)

Let's remember physical meaning these mathematical phrases.

The first equation (3.1) states that electrostatic the field can only be created by electric charges. In this equation - vector electrical displacement, ρ - volumetric density of electric charge.

The electric displacement vector flux through any closed surface is equal to the charge contained within that surface.

As experiment shows, the flux of the magnetic induction vector through a closed surface is always zero (3.2)

A comparison of equations (3.2) and (3.1) allows us to conclude that there are no magnetic charges in nature.

Equations (3.3) and (3.4) are of great interest and importance. Here we consider the circulation of electric voltage vectors ( ) and magnetic ( ) fields along a closed contour.

Equation (3.3) states that the alternating magnetic field ( ) is the source of the vortex electric field ( ).This is nothing more than a mathematical representation of the phenomenon of Faraday electromagnetic induction.

Equation (3.4) establishes the connection between the magnetic field and the alternating electric field. According to this equation, a magnetic field can be created not only by conduction current ( ), but also by an alternating electric field .

In these equations:

- electric displacement vector,

H- magnetic field strength,

E- electric field strength,

j- conduction current density,

μ - magnetic permeability of the medium,

ε is the dielectric constant of the medium.

Electromagnetic waves. Properties of electromagnetic waves

Last semester, completing our consideration of Maxwell's system of equations of classical electrodynamics, we established that joint decision the last two equations (about the circulation of vectors And ) leads to a differential wave equation.

So we got the wave equation of the “Y” wave:

. (3.6)

Electrical component y - waves propagate in the positive direction of the X axis with phase velocity

(3.7)

A similar equation describes the change in space and time of the magnetic field y - wave:

. (3.8)

Analyzing the results obtained, it is possible to formulate a number of properties inherent in electromagnetic waves.

1. A plane “y” wave is a linearly polarized transverse wave. Electrical voltage vectors ( ), magnetic ( ) field and wave phase velocity ( ) are mutually perpendicular and form a “right-handed” system (Fig. 3.1).

2. At each point in space the wave component H z is proportional to the electric field strength E y:

Here the “+” sign corresponds to a wave propagating in the positive direction of the X axis. The “-” sign corresponds to the negative one.

3. An electromagnetic wave moves along the X axis with phase velocity

Here
.

When an electromagnetic wave propagates in a vacuum (ε = 1, μ = 1), the phase velocity

Here the electrical constant ε 0 = 8.85 10 -12

magnetic constant μ 0 = 4π 10 -7

The coincidence of the speed of an electromagnetic wave in a vacuum with the speed of light was the first proof of the electromagnetic nature of light.

In a vacuum, the connection between the strength of the magnetic and electric fields in the wave is simplified.