The period of change in the energy of the magnetic field of the coil. Magnetic field energy

An inductor is an electronic component that is a screw or spiral structure made using an insulated conductor. The main property of an inductor, as the name implies, is inductance. Inductance is the property of converting the energy of an electric current into the energy of a magnetic field. The inductance value for a cylindrical or ring coil is

Where ψ is the flux linkage, µ0 = 4π*10-7 is the magnetic constant, N is the number of turns, S is the cross-sectional area of the coil.

Also, the inductor has such properties as small capacitance and low active resistance, and an ideal coil is completely devoid of them. The use of this electronic component is observed almost everywhere in electrical devices. The purposes of application are different:

Suppression of interference in the electrical circuit;
- smoothing the level of pulsations;
- accumulation of energy potential;
- limitation of variable frequency currents;
- construction of resonant oscillatory circuits;
- filtering frequencies in electrical signal circuits;
- formation of a magnetic field region;
- construction of delay lines, sensors, etc.

Magnetic field energy of an inductor

Electric current contributes to the accumulation of energy in the magnetic field of the coil. If the electricity supply is turned off, the accumulated energy will be returned to the electrical circuit. In this case, the voltage value in the coil circuit increases many times. The amount of stored energy in the magnetic field is approximately equal to the amount of work that must be obtained to ensure the appearance of the required current in the circuit. The value of the energy stored by the inductor can be calculated using the formula.

Reactance

When alternating current flows, the coil has, in addition to active, also reactance, which is found according to the formula

The formula shows that, unlike a capacitor, the reactance of a coil increases with increasing frequency; this property is used in frequency filters.

When constructing vector diagrams, it is important to remember that in a coil, voltage leads current by 90 degrees.

Coil quality factor

Another important property of a coil is its quality factor. The quality factor shows the ratio of the reactance of the coil to the active one.

The higher the quality factor of the coil, the closer it is to ideal, that is, it has only its main property - inductance.

Inductor designs

Structurally, inductors can be presented in different designs. For example, in the design of single-layer or multi-layer winding of the conductor. In this case, wire winding can be performed on dielectric frames of different shapes: round, square, rectangular. The production of frameless coils is often practiced. The method of manufacturing toroidal coils is widely used.

The inductance of the coil can be changed by adding a ferromagnetic core to the coil design. The introduction of cores affects interference suppression. Therefore, almost all chokes designed to suppress high-frequency interference, as a rule, have ferrodielectric cores made on the basis of ferrite, fluxtrol, ferroxon, carbonyl iron. Low-frequency interference is well smoothed out by coils on permalloy cores or electrical steel cores.

>> Magnetic field energy of current

§ 16 ENERGY OF THE MAGNETIC FIELD OF CURRENT

According to the law of conservation of energy, the energy of the magnetic field created by the current is equal to the energy that the current source (galvanic cell, generator at a power plant, etc.) must expend to create the current. When the circuit is opened, this energy transforms into other types of energy.

The fact that to create a current it is necessary to expend energy, i.e., it is necessary to do work, is explained by the fact that when the circuit is closed, when the current begins to increase, a vortex electric field appears in the conductor, acting against the electric field that is created in the conductor due to the source current In order for the current strength to become equal to /, the current source must do work against the forces of the vortex field. This work goes to increase the energy of the magnetic field of the current.

When the circuit is opened, the current disappears and the vortex field does positive work. The energy stored in the current is released. This is detected, for example, by a powerful spark that occurs when a circuit with high inductance is opened.

The energy of the magnetic field created by a current passing through a section of a circuit with inductance L is determined by the formula

The energy of the magnetic field is expressed here through the characteristic of the conductor L and the current strength in it /. But this same energy can also be expressed through the characteristics of the field. Calculations show that the energy density of the magnetic field (i.e., the energy per unit volume) is proportional to the square of the magnetic induction: , just as the energy density of the electric field is proportional to the square of the electric field strength.

Lesson content lesson notes supporting frame lesson presentation acceleration methods interactive technologies Practice tasks and exercises self-test workshops, trainings, cases, quests homework discussion questions rhetorical questions from students Illustrations audio, video clips and multimedia photographs, pictures, graphics, tables, diagrams, humor, anecdotes, jokes, comics, parables, sayings, crosswords, quotes Add-ons abstracts articles tricks for the curious cribs textbooks basic and additional dictionary of terms other Improving textbooks and lessonscorrecting errors in the textbook updating a fragment in a textbook, elements of innovation in the lesson, replacing outdated knowledge with new ones Only for teachers perfect lessons calendar plan for the year; methodological recommendations; discussion program Integrated Lessons

« Physics - 11th grade"

Self-induction.

If alternating current flows through the coil, then:
the magnetic flux passing through the coil varies with time,
and an induced emf occurs in the coil.
This phenomenon is called self-induction.

According to Lenz's rule, as the current increases, the intensity of the vortex electric field is directed against the current, i.e. the vortex field prevents the current from increasing.
When the current decreases, the intensity of the vortex electric field and the current are directed in the same way, i.e. the vortex field supports the current.

The phenomenon of self-induction is similar to the phenomenon of inertia in mechanics.

In mechanics:
Inertia causes a body to gradually acquire a certain speed under the influence of force.
The body cannot be instantly slowed down, no matter how great the braking force.

In electrodynamics:
When the circuit is closed due to self-induction, the current strength increases gradually.
When the circuit is opened, self-induction maintains the current for some time, despite the resistance of the circuit.

The phenomenon of self-induction plays a very important role in electrical and radio engineering.

Current magnetic field energy

According to the law of conservation of energy magnetic field energy, created by the current, is equal to the energy that the current source (for example, a galvanic cell) must expend to create the current.
When the circuit is opened, this energy transforms into other types of energy.

When closed circuit current increases.
A vortex electric field appears in the conductor, acting against the electric field created by the current source.
In order for the current strength to become equal to I, the current source must do work against the forces of the vortex field.
This work goes to increase the energy of the magnetic field of the current.

When opening circuit current disappears.
The vortex field does positive work.
The energy stored in the current is released.
This is detected, for example, by a powerful spark that occurs when a circuit with high inductance is opened.

The energy of the magnetic field created by a current passing through a section of a circuit with inductance L is determined by the formula

The magnetic field created by an electric current has an energy directly proportional to the square of the current.

The energy density of the magnetic field (i.e., the energy per unit volume) is proportional to the square of the magnetic induction: w m ~ V 2,
similar to how the energy density of the electric field is proportional to the square of the electric field strength w e ~ E 2.

Magnetic field energy.

A magnetic field- a force field acting on moving electric charges and on bodies with a magnetic moment, regardless of the state of their motion, the magnetic component of the electromagnetic field

A magnetic field can be created by the current of charged particles and/or the magnetic moments of electrons in atoms (and the magnetic moments of other particles, although to a noticeably lesser extent) (permanent magnets).

Magnetic field energy, created by the current in a closed circuit by inductance L, is equal to where I is the current strength in the circuit.

Magnetic field energy coil with inductance L created by current I is equal to

Magnetic field energy

A conductor through which electric current flows is always surrounded by a magnetic field, and the magnetic field appears and disappears along with the appearance and disappearance of the current. A magnetic field, like an electric field, is a carrier of energy. It is natural to assume that the energy of the magnetic field is equal to the work expended by the current to create this field.

L, by to which the current flows I. A magnetic flux is coupled to this circuit (see (126.1)) F =LI, I L d IA=I dФ =LI d I.

Because I=Bl/(m 0 mN) (see (119.2)) and B=m 0 mH(see (109.3)), then

(130.2)

Where Sl = V - solenoid volume.

(130.3)

IN from N linear, those. it applies only to para- and diamagnetic materials.

Electromagnetic field energy

Electromagnetic field energy- energy contained in the electromagnetic field [ source not specified 1754 days]. This also includes special cases of pure electric and pure magnetic fields.

Work done by an electric field to move a charge

The concept of work A (\displaystyle A) of the electric field E (\displaystyle E) to move the charge Q (\displaystyle Q) is introduced in full accordance with the definition of mechanical work:

A = ∫ F (x) d x = ∫ Q ⋅ E (x) d x = Q ⋅ U (\displaystyle A=\int F(x)\,dx=\int Q\cdot E(x)\,dx=Q \cdot U)

where U = ∫ E d x (\displaystyle U=\int E\,dx) is the potential difference (the term voltage is also used).

In many problems, continuous charge transfer is considered over a period of time between points with a given potential difference U (t) (\displaystyle U(t)), in which case the formula for the work should be rewritten as follows:

A = ∫ U (t) d Q = ∫ U (t) I (t) d t (\displaystyle A=\int U(t)\,dQ=\int U(t)I(t)\,dt)

where I (t) = d Q d t (\displaystyle I(t)=(dQ \over dt)) is the current strength.

Electric current power in the circuit

The power W (\displaystyle W) of the electric current for a section of the circuit is determined in the usual way, as the derivative of the work A (\displaystyle A) with respect to time, that is, by the expression:

W (t) = d A d t = U (t) ⋅ I (t) (\displaystyle W(t)=(\frac (dA)(dt))=U(t)\cdot I(t))

This is the most general expression for power in an electrical circuit.

Taking into account Ohm's law

U = I ⋅ R (\displaystyle U=I\cdot R)

the electrical power released at the resistance R (\displaystyle R) can be expressed in terms of current

W = I (t) 2 ⋅ R (\displaystyle W=I(t)^(2)\cdot R) ,

and through voltage:

W = U (t) 2 R (\displaystyle W=((U(t)^(2)) \over R))

Accordingly, work (heat released) is the integral of power over time:

A = ∫ W (t) d t = ∫ I (t) 2 ⋅ R d t = ∫ U (t) 2 R d t (\displaystyle A=\int W(t)\,dt=\int I(t)^( 2)\cdot R\,dt=\int ((U(t)^(2)) \over R)\,dt)

Energy of electric and magnetic fields

For electric and magnetic fields, their energy is proportional to the square of the field strength. Strictly speaking, the term “electromagnetic field energy” is not entirely correct. Instead, in physics they usually use the concept electromagnetic field energy density(at a certain point in space). The total energy of the field is equal to the integral of the energy density over the entire space.

The energy density of the electromagnetic field is the sum of the energy densities of the electric and magnetic fields.

In the SI system:

U = E ⋅ D 2 + B ⋅ H 2 (\displaystyle u=(\frac (\mathbf (E) \cdot \mathbf (D) )(2))+(\frac (\mathbf (B) \cdot \ mathbf (H) )(2)))

In vacuum (as well as in matter when considering microfields):

U = ε 0 E 2 2 + B 2 2 μ 0 = ε 0 E 2 + c 2 B 2 2 = E 2 / c 2 + B 2 2 μ 0 (\displaystyle u=(\varepsilon _(0)E^ (2) \over 2)+(B^(2) \over (2\mu _(0)))=\varepsilon _(0)(\frac (E^(2)+c^(2)B^ (2))(2))=(\frac (E^(2)/c^(2)+B^(2))(2\mu _(0))))

Where E- electric field strength, B- magnetic induction, D- electrical induction, H- magnetic field strength, With- the speed of light, ε 0 (\displaystyle \varepsilon _(0)) - the electrical constant and μ 0 (\displaystyle \mu _(0)) - the magnetic constant. Sometimes for the constants ε 0 (\displaystyle \varepsilon _(0)) and μ 0 (\displaystyle \mu _(0)) the terms dielectric constant and magnetic permeability of vacuum are used, which are extremely unfortunate and are now almost never used.

In the GHS system:

U = E ⋅ D + B ⋅ H 8 π (\displaystyle u=(\frac (\mathbf (E) \cdot \mathbf (D) +\mathbf (B) \cdot \mathbf (H) )(8\pi )))

Electromagnetic field energy in an oscillatory circuit

Electromagnetic field energy in an oscillatory circuit:

W = C U 2 2 + L I 2 2 (\displaystyle W=(\frac (CU^(2))(2))+(\frac (LI^(2))(2)))

U is the electrical voltage in the circuit, C is the electrical capacity of the capacitor, I is the current strength, L is the inductance of the coil or turn with current.

Electromagnetic field energy flows

Main article: Poynting vector

For an electromagnetic wave, the energy flux density is determined by the Poynting vector S (in the Russian scientific tradition - the Umov-Poynting vector).

In the SI system, the Poynting vector is equal to S = E × H (\displaystyle \mathbf (S) =\mathbf (E) \times \mathbf (H) ) (the vector product of the electric and magnetic field strengths) and is directed perpendicular to the vectors E and H. This naturally agrees with the transverse property of electromagnetic waves.

At the same time, the formula for the energy flux density can be generalized for the case of stationary electric and magnetic fields and has the same form: S = E × H (\displaystyle \mathbf (S) =\mathbf (E) \times \mathbf (H ) ) .

The fact of the existence of energy flows in constant electric and magnetic fields may look strange, but does not lead to any paradoxes; Moreover, such flows are detected in experiment.

Magnetic field energy

When the inductor coil is disconnected from the current source, an incandescent lamp connected parallel to the coil gives a short-term flash. The current in the circuit arises under the influence of self-induction emf. The source of energy released in the electrical circuit is the magnetic field of the coil.

The energy of the magnetic field of the inductor can be calculated in the following way. To simplify the calculation, consider the case when, after disconnecting the coil from the source, the current in the circuit decreases with time according to a linear law. In this case, the self-induction emf has a constant value equal to

where t is the time period during which the current in the circuit decreases from the initial value I to 0.

During time t, with a linear decrease in current strength from I to 0, an electric charge passes through the circuit:

therefore the work done by the electric current is

This work is done due to the energy of the magnetic field of the coil. The energy of the magnetic field of an inductor is equal to half the product of its inductance and the square of the current in it:

Maxwell's equation. Electromagnetic waves.

According to Maxwell's theory, an alternating magnetic field causes the appearance of an alternating vortex electric. field, which, in turn, causes the appearance of an alternating magnetic field, etc. In this way, electromagnetic disturbances propagate in space, i.e. an electromagnetic wave propagates. Basic properties of electromagnetic waves. 1. Electromagnetic wave – transverse. 2. The speed of electromagnetic waves in a vacuum is equal to v=c=3*108m/s and coincides with the speed of light. In a medium v=c/(), where  and  are the dielectric and magnetic permeabilities of the medium. 3. Electromagnetic waves carry energy. 4. Electromagnetic waves are reflected from conducting surfaces and refracted at the boundary of two dielectrics. 5. Electromagnetic waves exert pressure on bodies. 6. If an electromagnetic wave exerts pressure on bodies, i.e. gives them an impulse, therefore it also has an impulse. 7. Diffraction, interference and polarization of electromagnetic waves are observed.

MAxwell equationenia, fundamental equations of classical macroscopic electrodynamics, describing electromagnetic phenomena in an arbitrary environment. M. u. formulated by J.K. Maxwell in the 60s of the 19th century based on a generalization of the empirical laws of electrical and magnetic phenomena. Based on these laws and developing the fruitful idea of M. Faraday that interactions between electrically charged bodies are carried out through electromagnetic field, Maxwell created the theory of electromagnetic processes, mathematically expressed by M. u. The modern form of M. u. given by the German physicist G. Hertz and the English physicist O. Heaviside.

M. u. connect the quantities characterizing the electromagnetic field with its sources, that is, with the distribution of electric charges and currents in space. In a vacuum, the electromagnetic field is characterized by two vector quantities depending on spatial coordinates and time: the electric field strength E and magnetic induction IN. These quantities determine the forces acting from the field on charges and currents, the distribution of which in space is given by the charge density r (charge per unit volume) and current density j(charge transferred per unit time through a unit area perpendicular to the direction of movement of the charges). To describe electromagnetic processes in the material environment (matter), except for vectors E And IN, auxiliary vector quantities are introduced, depending on the state and properties of the medium: electrical induction D and magnetic field strength N.

M. u. allow you to determine the main characteristics of the field ( E, B, D And N) at every point in space at any time, if the field sources are known j and r as functions of coordinates and time. M. u. can be written in integral or differential form (they are given below in the absolute system of Gaussian units; see GHS system of units).

M. u. in integral form it is not the field vectors themselves that are determined from given charges and currents E, B, D, H at individual points in space, and some integral quantities depending on the distribution of these field characteristics: circulation vectors E And N along arbitrary closed contours and streams vectors D And B through arbitrary closed surfaces.

First M. u. is a generalization to variable fields of the empirical Ampere Law on the excitation of a magnetic field by electric currents. Maxwell hypothesized that the magnetic field is generated not only by currents flowing in conductors, but also by alternating electric fields in dielectrics or vacuum. A quantity proportional to the rate of change of the electric field over time was called displacement current by Maxwell. The displacement current excites a magnetic field according to the same law as the conduction current (this was later confirmed experimentally). The total current, equal to the sum of the conduction current and the displacement current, is always closed.

First M. u. has the form:

, (1, a)

that is, the circulation of the magnetic field strength vector along a closed loop L(sum of scalar products of the vector N at a given point on the contour for an infinitesimal segment dl circuit) is determined by the total current through an arbitrary surface S j n- projection of conduction current density j to the normal to the infinitesimal area ds, which is part of the surface S, is the projection of the displacement current density onto the same normal, and With= 3×1010 cm/sec - a constant equal to the speed of propagation of electromagnetic interactions in a vacuum.

Second M. u. is a mathematical formulation of Faraday's law of electromagnetic induction (see. Electromagnetic induction) is written as:

that is, the circulation of the electric field strength vector along a closed loop L(induction emf) is determined by the rate of change of the magnetic induction vector flux through the surface S, limited by this contour. Here B n- projection to the normal to the site ds magnetic induction vector IN; minus sign corresponds Lenz's rule to direct the induction current.

Third M. u. expresses experimental data on the absence of magnetic charges similar to electric ones (the magnetic field is generated only by currents):

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

Fourth M. u. (commonly called Gauss's theorem) is a generalization of the law of interaction of stationary electric charges - Pendant law:

, (1, g)

that is, the flow of the electrical induction vector through an arbitrary closed surface S determined by the electric charge located inside this surface (in the volume V, limited by a given surface).

If we assume that the electromagnetic field vectors ( E, B, D, H) are continuous functions of coordinates, then, considering the circulation of vectors N And E along infinitesimal contours and vector flows B And D Through surfaces that limit infinitely small volumes, it is possible to move from integral relations (1, a - d) to a system of differential equations that are valid at every point in space, that is, to obtain the differential form of mathematical equations. (usually more convenient for solving various problems):

rot ,

Here rot and div are rotor differential operators (see Vortex) And divergence, acting on vectors N, E, B And D. The physical meaning of equations (2) is the same as equations (1).

M. u. in form (1) or (2) do not form a complete closed system that allows one to calculate electromagnetic processes in the presence of a material environment. It is necessary to supplement them with relations connecting the vectors E, H, D, B And j, which are not independent. The connection between these vectors is determined by the properties of the medium and its state, and D And j are expressed through E, A B- through N:

D = D (E), B = B (N), j = j (E). (3)

These three equations are called equations of state, or material equations; they describe the electromagnetic properties of the medium and have a specific shape for each specific medium. In a vacuum Dº E And Bº N. The set of field equations (2) and state equations (3) form a complete system of equations.

Macroscopic M. at. describe the medium phenomenologically, without considering the complex mechanism of interaction of the electromagnetic field with charged particles of the medium. M. u. can be obtained from Lorentz - Maxwell equations for microscopic fields and certain ideas about the structure of matter by averaging microfields over small space-time intervals. In this way, both the basic field equations (2) and the specific form of the equations of state (3) are obtained, and the form of the field equations does not depend on the properties of the medium.

In the general case, the equations of state are very complex, since the vectors D, B And j at a given point in space at a given moment in time may depend on the fields E And N at all points in the environment at all previous times. In some environments, vectors D And B may be different from zero when E And H equal to zero ( ferroelectrics And ferromagnets). However, for most isotropic media, up to very significant fields, the equations of state have a simple linear form:

D= e E, B= m H, j= s E+ j page (4)

Here e ( x, y, z) - the dielectric constant, and m ( x, y, z) - magnetic permeability environments characterizing its electrical and magnetic properties, respectively (in the chosen system of units for vacuum e = m = 1); value s( x, y, z) is called specific electrical conductivity; j pp - the density of the so-called extraneous currents, that is, currents supported by any forces other than the forces of the electric field (for example, magnetic field, diffusion, etc.). In Maxwell's phenomenological theory, the macroscopic characteristics of the electromagnetic properties of the medium e, m and s must be found experimentally. In the microscopic Lorentz-Maxwell theory they can be calculated.

The permeabilities e and m actually determine the contribution to the electromagnetic field made by the so-called bound charges that are part of the electrically neutral atoms and molecules of the substance. The experimental determination of e, m, s allows one to calculate the electromagnetic field in a medium without solving the difficult auxiliary problem of the distribution of bound charges and their corresponding currents in the substance. Charge density r and current density j in M. u. are the densities of free charges and currents, and the auxiliary vectors N And D are introduced so that the circulation of the vector N was determined only by the movement of free charges, and the flow of the vector D- density of distribution of these charges in space.

If the electromagnetic field is considered in two adjacent media, then at their interface the field vectors may undergo discontinuities (jumps); in this case, equations (2) must be supplemented with boundary conditions:

[nH] 2 - [nH] 1 = ,

[nE] 2 - [nE] 1 = 0, (5)

(nD) 2 - (nD) 1 = 4ps,

(nB) 2 - (nB) 1 = 0.

Here j pov and s are the surface current and charge densities, square and parentheses are the vector and scalar products of vectors, respectively, n is the unit normal vector to the interface in the direction from the first medium to the second (1®2), and the indices refer to different sides of the interface.

The basic equations for the field (2) are linear, while the equations of state (3) can also be nonlinear. Typically, nonlinear effects are detected in sufficiently strong fields. In linear media [satisfying relations (4)] and, in particular, in vacuum, M. at. linear and thus turns out to be fair superposition principle: When fields overlap, they do not affect each other.

From M. u. a number of conservation laws follow. In particular, from equations (1, a) and (1, d) we can obtain the relation (the so-called continuity equation):

which is the law of conservation of electric charge: the total current flowing per unit time through any closed surface S, is equal to the change in charge inside the volume V, limited by this surface. If there is no current through the surface, then the charge in the volume remains unchanged.

From M. u. it follows that the electromagnetic field has energy and momentum (amount of motion). The energy density w (energy per unit volume of the field) is equal to:

Electromagnetic energy can travel in space. The energy flux density is determined by the so-called Poynting vector

The direction of the Poynting vector is perpendicular to as E, so N and coincides with the direction of propagation of electromagnetic energy, and its value is equal to the energy transferred per unit time through a unit surface perpendicular to the vector P. If there are no transformations of electromagnetic energy into other forms, then, according to mathematical equations, the change in energy in a certain volume per unit time is equal to the flow of electromagnetic energy through the surface bounding this volume. If heat is released inside a volume due to electromagnetic energy, then the law of conservation of energy is written in the form:

Where Q- the amount of heat released per unit time.

Electromagnetic field pulse density g(momentum per unit volume of the field) is related to the energy flux density by the relation:

The existence of an electromagnetic field pulse was first discovered experimentally in the experiments of P.N. Lebedeva on measuring the pressure of light (1899).

As can be seen from (7), (8) and (10), the electromagnetic field always has energy, and the energy flow and electromagnetic impulse are nonzero only in the case when both electric and magnetic fields exist simultaneously (and these fields are not parallel to each other ).

M. u. lead to the fundamental conclusion about the finiteness of the propagation speed of electromagnetic interactions (equal to With= 3×1010 cm/sec). This means that when the charge or current density changes at a certain point in space, the electromagnetic field generated by them at the observation point changes not at the same moment in time, but after a time t = R/c, Where R- distance from the current element or charge to the observation point. Due to the finite speed of propagation of electromagnetic interactions, the existence of electromagnetic waves, a special case of which (as Maxwell first showed) are light waves.

Electromagnetic phenomena occur in the same way in all inertial reference systems, that is, they satisfy the principle of relativity. In accordance with this, M. u. do not change their shape when moving from one inertial frame of reference to another (relativistically invariant). The fulfillment of the principle of relativity for electromagnetic processes turned out to be incompatible with classical ideas about space and time, required a revision of these ideas and led to the creation of a special theory of relativity (A. Einstein, 1905; cm. Relativity theory). Form M. u. remains unchanged upon transition to a new inertial reference system, if space, coordinates and time, field vectors E, H, B, D, current density j and charge density r change in accordance with Lorentz transformations(expressing new, relativistic ideas about space and time). Relativistically invariant form of M. at. emphasizes the fact that the electric and magnetic fields form a single whole.

M. u. describe a huge range of phenomena. They form the basis of electrical engineering and radio engineering and play a vital role in the development of such current areas of modern physics as physics plasma and the problem of the governed thermonuclear reactions, magnetic hydrodynamics, nonlinear optics, design charged particle accelerators, astrophysics, etc. M.u. are inapplicable only at high frequencies of electromagnetic waves, when quantum effects become significant, that is, when the energy of individual quanta of the electromagnetic field - photons - is high and a relatively small number of photons are involved in the processes.

§ 130. Magnetic field energy

Consider a circuit with inductance L, through which current flows I. A magnetic flux is coupled to this circuit (see (126.1)) Ф= LI, and when the current changes by d I magnetic flux changes by dФ= L d I. However, to change the magnetic flux by an amount dФ (see § 121), it is necessary to perform work d A=I dФ= LI d I. Then the work to create magnetic flux Ф will be equal to

Therefore, the energy of the magnetic field associated with the circuit is

W=LI2 /2. (130.1)

The study of the properties of alternating magnetic fields, in particular the propagation of electromagnetic waves, provided evidence that the energy of the magnetic field is localized in space. This corresponds to the concepts of field theory.

The magnetic field energy can be

set as a function of the quantities characterizing this field in the surrounding space. To do this, consider a special case - a uniform magnetic field inside a long solenoid. Substituting expression (126.2) into formula (130.1), we obtain

Because I=Bl/ ( 0 N) (see (119.2)) and B= 0  H(see (109.3)), then

Where Sl=V - solenoid volume.

The magnetic field of the solenoid is uniform and concentrated inside it, therefore the energy (see (130.2)) is contained in the volume of the solenoid and is distributed in it with a constant bulk density

Expression (130.3) for the volumetric energy density of a magnetic field has a form similar to formula (95.8) for the volumetric energy density of an electrostatic field, with the difference that electrical quantities are replaced by magnetic ones. Formula (130.3) was derived for a homogeneous field, but it is also valid for inhomogeneous fields. Expression (130.3) is valid only for media for which the dependence IN from N linear, that is, it applies only to para- and diamagnets (see § 132).

Control questions

What is the phenomenon of electromagnetic induction? Analyze Faraday's experiments.

What causes the emf to occur? induction in a closed conducting loop? Why and how does emf depend? induction occurring in the circuit?

Why is it better to use a closed conductor to detect induced current?

in the form of a coil, and not in the form of one turn of wire?

Formulate Lenz's rule, illustrating it with examples.

When changing the flux of magnetic induction in a conducting circuit, does an emf always arise in it? induction? induced current?

Does an induced current arise in a conducting frame moving forward in a uniform magnetic field?

Show that Faraday's law is a consequence of the law of conservation of energy.

What is the nature of emf. electromagnetic induction?

Derive an expression for emf. induction in a flat frame rotating uniformly in a uniform magnetic field. How can it be increased?

What are eddy currents? Are they harmful or beneficial?

Why aren't transformer cores made solid?

What are the phenomena of self-induction and mutual induction? Calculate the emf. induction

for both cases,

What is the physical meaning of relaxation time = L/R Prove that it has

dimension of time.

Give the relationship between the currents in the primary and secondary windings of the step-up transformer.

When the e.m.f. Is there more self-induction when the DC circuit is closed or opened?

What physical quantity is expressed in henry? Define Henry.

What is the physical meaning of circuit inductance? mutual inductance of two circuits? What do they depend on?

Write and analyze expressions for the volumetric energy density of electrostatic and magnetic fields. What is the volumetric energy density of the electromagnetic field?

The magnetic field strength doubled. How has the volumetric energy density of the magnetic field changed?

Tasks

15.1. A ring of aluminum wire (=26 nOhm m) is placed in a magnetic field perpendicular to the magnetic induction lines. Ring diameter 20 cm, wire diameter 1 mm. Determine the rate of change of the magnetic field if the current in the ring is 0.5 A.

15.2. In a uniform magnetic field, the induction of which is 0.5 T, a coil containing 200 turns, tightly adjacent to each other, rotates uniformly with a frequency of 300 min-1. The cross-sectional area of the coil is 100 cm2. The axis of rotation is perpendicular to the axis of the coil and the direction of the magnetic field. Determine the maximum emf induced in the coil. .

15.3. Determine how many turns of wire, closely adjacent to each other, with a diameter of 0.3 mm with insulation of negligible thickness, must be wound on a cardboard cylinder with a diameter of 1 cm to obtain a single-layer coil with an inductance of 1 mH.

15.4. Determine how long it will take for the circuit current to reach 0.98 of the limit value if the current source is connected to a coil with a resistance of 10 Ohms and an inductance of 0.4 H.

15.5. Two solenoids (the inductance of one L 1 =0.36 Hn, second L 2 = 0.64 H) of equal length and almost equal cross-section are inserted into one another. Determine the mutual inductance of the solenoids.

15.6. Autotransformer that reduces voltage from U 1 =5.5 kV up to U 2 =220 V, contains in the primary winding N 1 = 1500 turns. Secondary winding resistance R 2 =2 Ohm. External circuit resistance (in low voltage network) R=13 Ohm. Neglecting the resistance of the primary winding, determine the number of turns in the secondary winding of the transformer.

37 Magnetic field energy

Consider a circuit with inductance L, By to which the current flows I. A magnetic flux is coupled to this circuit (see (126.1)) F = LI, and when the current changes by d I magnetic flux changes by dФ= L d I. However, to change the magnetic flux by an amount dФ (see § 121), it is necessary to perform work d A=I dФ = LI d I. Then the work to create magnetic flux Ф will be equal to

Therefore, the energy of the magnetic field associated with the circuit is

(130.1)

The energy of a magnetic field can be represented as a function of the quantities characterizing this field in the surrounding space. To do this, consider a special case - a uniform magnetic field inside a long solenoid. Substituting expression (126.2) into formula (130.1), we obtain

Because I= Bl/ ( 0  N) (see (119.2)) and B= 0  H(see (109.3)), then

130.2)

Where Sl= V- solenoid volume.

(130.3)

Expression (130.3) for the volumetric energy density of a magnetic field has a form similar to formula (95.8) for the volumetric energy density of an electrostatic field, with the difference that electrical quantities are replaced in it by magnetic ones. Formula (130.3) was derived for a homogeneous field, but it is also valid for inhomogeneous fields. Expression (130.3) is valid only for media for which the dependence IN from N linear, those. it applies only to para- and diamagnets (see § 132).

38. Magnetic moments of electrons and atoms

When considering the effect of a magnetic field on conductors carrying current and on moving charges, we were not interested in the processes occurring in matter. The properties of the medium were taken into account formally using magnetic permeability  . In order to understand the magnetic properties of media and their influence on magnetic induction, it is necessary to consider the effect of a magnetic field on the atoms and molecules of a substance.

Experience shows that all substances placed in a magnetic field are magnetized. Let us consider the cause of this phenomenon from the point of view of the structure of atoms and molecules, basing it on Ampere’s hypothesis (see § 109), according to which in any body there are microscopic currents caused by the movement of electrons in atoms and molecules.

For a qualitative explanation of magnetic phenomena, with a sufficient approximation, we can assume that the electron moves in an atom in circular orbits. An electron moving in one of these orbits is equivalent to a circular current, so it has orbital magnetic moment(see (109.2)) p m = ISn, whose modulus

(131.1)

Where I= e - current strength,  - frequency of electron rotation in orbit, S- orbital area. If the electron moves clockwise (Fig. 187), then the current is directed counterclockwise and the vector R m (in accordance with the right-hand screw rule) is directed perpendicular to the electron orbital plane, as indicated in the figure.

On the other hand, an electron moving in orbit has a mechanical angular momentum L e, whose modulus, according to (19.1),

(131.2)

Where v = 2 ,  r 2 = S. Vector L e(its direction is also determined by the right screw rule) is called orbital mechanical momentum of the electron.

From Fig. 187 it follows that the directions R m and L e, are opposite, therefore, taking into account expressions (131.1) and (131.2), we obtain

(131.3)

where is the value

(131.4)

called gyromagnetic ratio of orbital moments(it is generally accepted to write with a “–” sign, indicating that the directions of the moments are opposite). This ratio, determined by the universal constants, is the same for any orbit, although for different orbits the values v And r are different. Formula (131.4) was derived for a circular orbit, but it is also valid for elliptical orbits.

The experimental determination of the gyromagnetic ratio was carried out in the experiments of Einstein and de Haas* (1915), who observed the rotation of an iron rod freely suspended on a thin quartz thread when it was magnetized in an external magnetic field (alternating current was passed through the solenoid winding with a frequency equal to the frequency of torsional oscillations of the rod) . When studying forced torsional vibrations of the rod, the gyromagnetic ratio was determined, which turned out to be equal to – (e/ m). Thus, the sign of the carriers responsible for molecular currents coincided with the sign of the electron charge, and the gyromagnetic ratio turned out to be twice as large as the previously introduced value g(see (131.4)). To explain this result, which was of great importance for the further development of physics, it was assumed and subsequently proven that, in addition to orbital moments (see (131.1) and (131.2)), the electron has own mechanical angular momentum L es, called spin. It was believed that spin was due to the rotation of the electron around its axis, which led to a number of contradictions. It has now been established that spin is an integral property of the electron, like its charge and mass. Spin the electron L es, corresponds intrinsic (cellular) magnetic moment R ms, proportional L es and directed in the opposite direction:

(131.5)

*IN. I. de Haas (1878-1960) - Dutch physicist.

Magnitude g s called gyromagnetic ratio of spin moments.

Projection of the intrinsic magnetic moment onto the direction of the vector IN can only take one of the following two values:

Where ħ= h/ (2) (h- Planck's constant),  b- Bohr magneton, which is a unit of the magnetic moment of an electron.

In the general case, the magnetic moment of an electron is the sum of the orbital and spin magnetic moments. The magnetic moment of an atom, therefore, consists of the magnetic moments of the electrons included in its composition and the magnetic moment of the nucleus (determined by the magnetic moments of the protons and neutrons entering the nucleus). However, the magnetic moments of nuclei are thousands of times smaller than the magnetic moments of electrons, so they are neglected. Thus, the total magnetic moment of an atom (molecule) p a is equal to the vector sum of the magnetic moments (orbital and spin) of the electrons entering the atom (molecule):

(131.6)

Let us once again draw attention to the fact that when considering the magnetic moments of electrons and atoms, we used the classical theory, without taking into account the restrictions imposed on the movement of electrons by the laws of quantum mechanics. However, this does not contradict the results obtained, since for the further explanation of the magnetization of substances it is only essential that the atoms have magnetic moments.

What is the magnetic field energy of a current-carrying coil?

Almagul"

ENERGY OF THE MAGNETIC FIELD OF CURRENT

Around a current-carrying conductor there is a magnetic field that has energy.
Where does it come from? The current source included in the electric chain has a reserve of energy.
At the moment of electrical closure. The current source circuit expends part of its energy to overcome the effect of the self-inductive emf that arises. This part of the energy, called the current’s own energy, goes to the formation of a magnetic field.

The energy of the magnetic field is equal to the intrinsic energy of the current.
The self-energy of the current is numerically equal to the work that the current source must do to overcome the self-induction emf in order to create a current in the circuit.

Loop inductance.

Consider a closed circuit through which current flows. It creates a magnetic field. Its value is proportional to the current.

where is the inductance of the circuit. The unit of inductance is henry

Communication with other units . If the current in the circuit changes, the magnetic field created by the current will change. Consequently, the magnetic flux coupled to the circuit will change. According to Faraday's law, an induced emf will occur in the circuit. The occurrence of an EMF in a circuit when the current strength changes in it is called self-induction. For example, let's find the inductance of the solenoid. We found that the magnetic flux through the solenoid (flux linkage) is equal to

(2)

Comparing Eq. (1) and (2) we find

In general, the inductance of a circuit depends on its shape, size and magnetic permeability of the environment in which the circuit is located. There is an analogy between inductance (connects current and magnetic flux) and capacitance (connects charge and electric field strength).

Mutual induction.

Let's consider two stationary contours located close to each other. Current flows through circuit 1. It creates a magnetic field that permeates circuit 2. The magnetic flux through circuit 2, which is proportional to the current:

(1)
- proportionality coefficient. When the current changes, the magnetic field and magnetic flux change. This leads to the appearance of EMF in the second circuit.

By analogy, we consider the case when the current flows through the second circuit. It creates a magnetic flux that permeates circuit 1.

When the current changes in circuit 1, an emf is induced

Experience shows that. The phenomenon of the occurrence of induced emf in one of the circuits when the current strength changes in the other is called mutual induction. - mutual inductance of the circuits. It depends on the shape, size and location of the circuits, as well as on the magnetic permeability of the medium.

Magnetic field energy.

Previously, it was found that the elementary work when moving a conductor with current in a magnetic field is equal to: . This expression applies to a current-carrying circuit. The current creates a magnetic field, it permeates the circuit. Magnetic flux coupled to the circuit. When the current in the circuit changes, the magnetic flux also changes. In this case, elementary work is performed. Work to create magnetic flux

This is the energy of the magnetic field associated with the circuit.

Let's consider a solenoid. Using the expression for the solenoid inductance, expressing the current in terms of the magnetic field induction and taking into account the coupling, we can obtain an expression for the total energy of the solenoid magnetic field.

Solenoid volume. At the same time, we took into account that the magnetic field inside the solenoid is uniform. Then the magnetic field energy per unit volume or magnetic field energy density is equal to:

Transformer

The phenomenon of mutual induction underlies the operation of transformers. These are devices designed to lower or increase the voltage in the network. Device diagram

There are two coils connected to each other by a magnetic circuit or a common core. The number of turns in the first and second coils is equal to and , respectively. An alternating electric current is passed through one of the coils. This current creates a magnetic field that is almost entirely concentrated in the core. It penetrates the turns of the winding of the second coil. If a source with EMF is connected to the winding of coil 1, then the current in the winding is determined according to Ohm’s law, taking into account the self-inductive EMF