4 Maxwell's equation says that. Maxwell's equations

Maxwell's introduction of the concept of displacement current led to the completion of the macroscopic theory he created electromagnetic field, which allows us to explain from a unified point of view not only electrical and magnetic phenomena, but also to predict new ones, the existence of which was subsequently confirmed.

Maxwell's theory is based on 4 equations:

1. The electric field can be either potential or vortex, so the strength of the resulting field is equal to:

This equation shows that magnetic fields can be excited either by moving charges (electric currents) or by alternating electric fields.

3. Gauss's theorem for the field:

We get

So, the complete system of Maxwell’s equations in integral form:

The quantities included in Maxwell's equations are not independent and there is a connection between them.

For isotropic, non-ferroelectric and non-ferromagnetic media, we write the connection formulas:

where is the electrical constant, is the magnetic constant,

Dielectric constant of the medium, m - magnetic permeability of the medium,

r - specific electrical resistance, - specific electrical conductivity.

From Maxwell's equations it follows that What:

source electric field can be either electric charges, or time-varying magnetic fields, which can be excited either by moving electric charges (currents) or by alternating electric fields.

Maxwell's equations are not symmetrical with respect to electric and magnetic fields. This is due to the fact that magnetic charges do not exist in nature.

If and (stationary fields), then Maxwell’s equations take the following form:

Sources of electrical stationary field are only electric charges, sources of stationary magnetic field are only conduction currents .

Electric and magnetic field in in this case independent of each other, which makes it possible to study the constant electric and magnetic fields separately.

Differential form of writing Maxwell's equations:

Integral form writing Maxwell's equations is more general if there are discontinuity surfaces. The differential form of writing Maxwell's equation assumes that all quantities in space and time change continuously.

Maxwell's equations are the most general equations for electric and magnetic fields in quiescent media. They play the same role in the doctrine of electromagnetism. important role, like Newton's laws in mechanics. From Maxwell's equations it follows that an alternating magnetic field is always associated with an alternating electric field, and an 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.

Properties of Maxwell's equations

Maxwell's equations are linear. 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 j. The property of linearity of Maxwell's equations is associated with the principle of superposition; if any two fields satisfy Maxwell's equations, then this also applies to the sum of these fields.

Maxwell's equations contain continuity equations expressing the law of conservation of electric charge. To obtain the continuity equation, it is necessary to take the divergence from both sides of the first of Maxwell’s equations in differential form:

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 form of Maxwell's equations when passing from one inertial system reference to another does not change, but the quantities included in them are converted according to certain rules. Those. Maxwell's equations are correct relativistic equations, unlike, for example, Newton's equations of mechanics.

Maxwell's equations are asymmetric with respect to electric and magnetic fields. This is due to the fact that electric charges exist in nature, and magnetic charges No.

From Maxwell's equations it follows important conclusion about the existence of a fundamentally new phenomenon: the electromagnetic field is capable of existing independently - without electrical charges and currents. Moreover, its change necessarily has a wave character. Fields of this kind are called electromagnetic waves. In a vacuum they always propagate at a speed equal speed Sveta. Maxwell's theory predicted the existence of electromagnetic waves and made it possible to establish all their basic properties.

In the case of stationary (that is, time-invariant) electric and magnetic fields, the origin of which is associated with stationary charges for the electric field and with stationary currents for the magnetic field, these fields are independent of each other, which allows us to consider them separately from each other.

Maxwell's equations is a system of equations that describe the nature of the origin and properties of electric and magnetic fields.

Maxwell's equations for stationary fields:

Thus, Maxwell's equations for stationary fields:

I.; II. ;

III.; IV. .

Vector characteristics of the electrostatic field And are related to each other by the following relationship:

,

Where – electrical constant,  – dielectric constant of the medium.

Vector characteristics of the magnetic field And are related to each other by the following relationship:

,

Where – magnetic constant, – magnetic permeability of the medium.

Topic 8. Maxwell's equations for the electromagnetic field

According to Maxwell's theories for the electromagnetic field in the case of non-stationary (that is, time-varying) electric and magnetic fields, the sources of the electric field can be either electric charges or a time-varying magnetic field, and the sources of the magnetic field can be either moving electric charges (electric currents) or an alternating electric field.

Unlike stationary fields, alternating electric and magnetic fields are not independent of each other and are considered as an electromagnetic field.

Maxwell's equations, as a system of equations describing the nature of the origin and properties of electric and magnetic fields in case electromagnetic field has the form:

I.
, that is, the circulation of the electric field strength vector is determined by the rate of change of the magnetic field induction vector ( rate of change of the induction vector ).

This equation shows that the sources of the electric field can be not only electric charges, but also time-varying magnetic fields.

II.
, that is, the vector flow electrical displacement through an arbitrary closed surface S, is equal algebraic sum charges contained inside the volume V, bounded by a given closed surface S ( - volumetric charge density).

III.
, that is, the circulation of the tension vector along an arbitrary closed contour L determined by total current I full piercing the surface S, limited by this contour L.

total current I full, consisting of conduction current I And bias current I cm., that is I full = I + I cm. .

Total conduction current I defined in general case through surface current density j (
) integration, that is

.

Bias current I cm piercing the surface S, is defined in general

case through surface bias current density
(
) integration, that is:
.

The concept of “displacement current” introduced by Maxwell, the magnitude of which is determined by the rate of change of the electric displacement vector , that is, the value , shows that magnetic fields can be excited not only by moving charges (electric conduction currents), but also by alternating electric fields.

IV.
, that is, the flux of the induction vector magnetic field through an arbitrary closed surface S equal to zero.

Maxwell's theory is based on the four equations discussed above:

1. The electric field can be either potential ( EQ), and vortex ( EB), therefore the total field strength E=EQ +EB. Since the circulation of the vector EQ is equal to zero (see (137.3)), and the circulation of the vector EB is determined by expression (137.2), 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(see (138.4)):

This equation shows that magnetic fields can be excited either by moving charges (electric currents) or by alternating electric fields.

3. Gauss's theorem for the field D(see (89.3)):

If the charge is distributed inside a closed surface continuously with bulk density r, then formula (139.1) will be written in the form

4. Gauss's theorem for the field IN(see (120.3)):

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 (isotropic non-ferroelectric and non-ferromagnetic media):

Where e 0 and m 0 - electric and magnetic constants, respectively, e And m- dielectric and magnetic permeability, respectively, g - conductivity substances.

From Maxwell's equations it follows that the sources of the electric field can be either electric charges or time-varying magnetic fields, and magnetic fields can be excited either by moving electric charges (electric currents) or by alternating electric fields. 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.

For stationary fields (E= const and B= const ) Maxwell's equations will take the form

those. In this case, the sources of the electric field are only electric charges, the sources of the magnetic field 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.

Using the Stokes and Gauss theorems known from vector analysis

one can imagine a complete system of Maxwell's equations in differential form(characterizing the field at each point in space):

If charges and currents are distributed continuously in space, then both forms of Maxwell's equations - integral and differential - are equivalent. However, if there are discontinuity surfaces - surfaces on which the properties of the medium or fields change abruptly, then the integral form of the equations is more general.

Maxwell's equations in differential form assume that all quantities in space and time vary continuously. To achieve mathematical equivalence of both forms of Maxwell's equations, the differential form is supplemented boundary conditions, which the electromagnetic field at the interface between two media must satisfy. The integral form of Maxwell's equations contains these conditions. These have been discussed before:

(the first and last equations correspond to cases when there is no free charges, no conduction currents).

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 electric field generated by it, and an 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.

Bias current or absorption current- a value directly proportional to the rate of change in electrical induction. This concept is used in classical electrodynamics

Introduced by J.C. Maxwell when constructing the theory of the electromagnetic field.

The introduction of a displacement current made it possible to eliminate the contradiction in the Ampere formula for the circulation of the magnetic field, which, after adding the displacement current, became consistent and constituted the last equation, which made it possible to correctly close the system of equations of (classical) electrodynamics.

Strictly speaking, the bias current is not electric shock, but is measured in the same units as electric current.

coefficient) is called the flow of the vector of rapidity of change of the electric field through a certain surface:

(SI)

Bias current. To generalize the equations of the electromagnetic field in vacuum to variable fields it is necessary to change only one of the previously written equations (see sections 3.4, 3.12); three equations turn out to be true in the general case. However, the law of total current for a magnetic field in the case of alternating fields and currents turns out to be incorrect. In accordance with this law, the current must be the same for any two surfaces stretched along the contour; if the charge in the volume between selected surfaces changes, then this statement conflicts with the law of conservation of charge. For example, when charging a capacitor (Fig. 45), the current through one of the indicated surfaces is equal and through the other (passing between the plates) - zero. To remove this contradiction, Maxwell introduced a displacement current into this equation, proportional to speed electric field changes:

In a dielectric medium, the expression for the displacement current takes the form:

The first term represents the displacement current density in vacuum, the second - real current, caused by the movement of bound charges when polarization changes. The displacement current through the surface is equal to where Ф is the vector flux through the surface. The introduction of a bias current removes the contradiction with the law of conservation of charge. For example, when charging flat capacitor displacement current through the surface passing between the plates, equal to current along the supply wires.

Maxwell's system of equations in vacuum. After introducing the displacement current, the system of Maxwell’s equations in differential form takes the form:

Maxwell's system of equations in integral form:

We also present a representation of Maxwell’s equations in differential form in the CGS system:

The charge and current densities are related by the relation

expressing the law of conservation of charge (this equation is a consequence of Maxwell’s equations).

Maxwell's equations in a medium have the form: differential form integral form

and serve to determine four quantities. To Maxwell's equations, in the medium, it is necessary to add material equations of connection between, characterizing electrical and magnetic properties environment. For isotropic linear media, these equations have the form:

From Maxwell's equations one can obtain boundary conditions for (see sections 3.6, 3.13).

Law of conservation of energy for the electromagnetic field.

From Maxwell's equations we can derive the following equation for any volume V bounded by a surface

The first term describes the change in the energy of the electromagnetic field in the volume under consideration. It can be seen that, in the general case, the energy density of the electromagnetic field turns out to be true formulas, obtained earlier for constant electric and magnetic fields. The second term represents the work of the field on the particles in the volume under consideration. Finally, the third term describes the flow of electromagnetic energy through the closed surface enclosing the volume. The energy flux density at a given point in space (Poynting vector) is determined by the vectors E and B at the same point:

The last expression is also valid for the flux density of electromagnetic energy in matter. The energy density in the medium has the form:

Example 1. Consider charging a flat capacitor with round plates located at a distance. The rate of change of energy in a cylinder of radius ( smaller sizes plates) is equal

We find the magnetic field strength from Maxwell's second equation: (on the right is the displacement current). We find that the rate of energy flow through lateral surface cylinder: equal to the rate of change of energy in the volume.

Relativistic properties of fields. When moving from one inertial reference system to another, both the sources of the electromagnetic field (charge and current densities) and the fields themselves change, but Maxwell’s equations retain their form. The simplest conversion formulas for sources are the density of a moving charge). If we denote the charge density in ISO, in which then, taking into account the reduction in longitudinal dimensions (see Section 1.11), we obtain

Comparing with the -vector of energy-momentum, we see that they form a -vector, i.e. are transformed through each other in the same way as according to the Lorentz transformation formulas. Knowing how field sources are converted, you can find formulas for converting E, B. They look like this:

Here is the speed of the reference frame K relative to the frame K, the transformations are written for field components parallel and perpendicular. The invariants of these transformations are scalar quantities

With c, the field conversion formulas take the following simplified form:

Example 2. Magnetic field of a nonrelativistic particle. Let us consider a particle that moves relative to ISO K with a constant relativistic speed V. In the ISO associated with a moving particle, there is only an electric field. To go to ISO K, you need to write the formulas

transformations Taking into account that in the nonrelativistic limit the lengths of the segments do not change, we obtain (for the moment when the particle passes through the origin of coordinates in K):

When deriving these formulas, we used the equality

Example 3. Polarization of a dielectric when moving in a magnetic field. When a dielectric moves at a nonrelativistic speed perpendicular to the magnetic field induction lines, its polarization occurs. In an IFR associated with a dielectric, there is a transverse electric field. The nature of the polarization of a dielectric depends on its shape.

Example 4. Electric field of a relativistic particle. Let's consider a particle that moves relative to the ISO K with a constant relativistic speed V. In the ISO K associated with the moving particle, there is only an electric field. To transfer to the ISO K, one should use the transformation formulas (92) with We write the answer for the moment of time when the particle is in the ISO K passes through the origin of coordinates, for a point lying in the plane. When moving from coordinates to coordinates, it is necessary to take into account that (the coordinates of the point are measured in K simultaneously with the passage of the particle through the origin of coordinates). As a result we get

It can be seen that vector E is collinear to vector However, at the same distance from the charge, the field at a point located on the line of its motion is less than at a point located perpendicular to the speed. The magnetic field at the same point is determined by the expression:

Note that the considered electric field is not potential.

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 - electric displacement vector, ρ - volumetric charge density.

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.

1. 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 wave equation"Y" waves:

. (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. 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 relationship between the strength of the magnetic and electric fields in the wave is simplified.

.

When an electromagnetic wave propagates in a dielectric medium (μ = 1)
And
.