The magnetic energy of the coil.

When the key is opened K the lamp flashes brightly

(8)

For coil
,
,


,

Bulk energy density

, . (9)

The total energy density of the electric and magnetic (electromagnetic) fields

(10)

Chapter 1. Electrodynamics Magnetic field

self induction is an important special case of electromagnetic induction, when a changing magnetic flux, causing an EMF of induction, is created by a current in the circuit itself. If the current in the circuit under consideration changes for some reason, then the magnetic field of this current changes, and, consequently, the own magnetic flux penetrating the circuit. In the circuit, an EMF of self-induction occurs, which, according to the Lenz rule, prevents a change in the current in the circuit.

Own magnetic fluxΦ, penetrating a circuit or coil with current, is proportional to the strength of the current I:

As an example, let's calculate the inductance of a long solenoid having N turns, cross-sectional area S and length l. The magnetic field of the solenoid is determined by the formula ( see § 1.17)

Therefore, the inductance of the solenoid is

Where V = Sl is the volume of the solenoid in which the magnetic field is concentrated. The result obtained does not take into account edge effects, so it is approximately valid only for sufficiently long coils. If the solenoid is filled with a substance with magnetic permeabilityμ, then at a given current I the magnetic field induction increases in absolute value by μ times ( see § 1.17); therefore, the inductance of the core coil also increases by a factor of μ:

The EMF of self-induction is directly proportional to the inductance of the coil and the rate of change in the current strength in it.

The magnetic field has energy. Just as a charged capacitor has a supply of electrical energy, a coil with current flowing through its coils has a supply of magnetic energy. If you turn on an electric lamp in parallel with a coil with a large inductance in a DC electrical circuit, then when the key is opened, a short flash of the lamp is observed (Fig. 1.21.1). The current in the circuit arises under the action of self-induction EMF. The source of energy released in this case in the electrical circuit is the magnetic field of the coil.

From the law of conservation of energy it follows that all the energy stored in the coil will be released in the form of Joule heat. If denoted by R impedance of the circuit, then in time Δ t the amount of heat released Δ Q = I 2 RΔ t.

The current in the circuit is

In this expression, Δ I < 0; ток в цепи постепенно убывает от первоначального значения I 0 to zero. The total amount of heat released in the circuit can be obtained by performing an integration operation ranging from I 0 to 0. This gives

Thus the energy W m magnetic field coil with inductance L generated by current I, is equal to

Where V is the volume of the solenoid. This expression shows that the magnetic energy is not localized in the turns of the coil through which the current flows, but is dispersed throughout the volume in which the magnetic field is created. Physical quantity

equal to the energy of the magnetic field per unit volume is called volume density of magnetic energy . J. Maxwell showed that the expression for the volume density of magnetic energy, derived here for the case of a long solenoid, is valid for any magnetic fields.

And on bodies with a magnetic moment, regardless of the state of their movement, the magnetic component of the electromagnetic field.

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

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

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

A conductor through which an 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 that is expended by the current to create this field.

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

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

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

The energy of the 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/(m 0 mN) (see (119.2)) and B = m 0 mH(see (109.3)), then

(130.2)

Where Sl = V - solenoid volume.

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

(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 electric 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 H linear, those. it only applies to

Consider the case we have already discussed (Figure 5.6).

First close the solenoid L to the EMF source, current will flow in it. Then at the moment of time we switch the key to position 2 - we close the solenoid to resistance R. Decreasing current will flow in the circuit I. In this case, the work will be done: , or

Let us express the energy of the magnetic field in terms of the parameters of the magnetic field. For solenoid:

.

; from here

Let us substitute these values ​​into the formula (5.5.3):

but because , That

The energy of a uniform magnetic field in a long solenoid can be calculated from the formula

The magnetic field energy density in a solenoid with a core will be the sum of the field energy in vacuum and in the core magnet:

, hence .

Because in a vacuum, we have

Control questions

1. What experiments of Faraday formed the basis for the discovery of the phenomenon of electromagnetic induction?

2. What is the cause of the induction EMF in a closed conducting circuit? On what and how does the induction emf that occurs in the circuit depend?

3. What is the phenomenon of electromagnetic induction?

4. Why is it better to use a closed conductor in the form of a coil, and not in the form of a single turn of wire, to detect induction current?

5. Formulate Lenz's rule, illustrating it with examples.

6. How is the induction current directed?

7. Does an EMF of induction always appear in a conducting circuit when the magnetic induction changes? induction current?

8. What is the EMF of the circuit induction?

9. Formulate Ohm's law for the circuit.

10. How is the direction of the induction current and the direction of the rate of change of the flux of magnetic induction related?

11. Formulate Faraday's law for a current-carrying circuit consisting of one or more turns.

12. Does an induction current occur in a conducting loop that is moving forward in a uniform magnetic field?

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

14. What is the nature of the EMF of electromagnetic induction?

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

16. What are eddy currents? Are they harmful or helpful?

17. Why are the cores of transformers not made solid?

18. What phenomenon is called the skin effect?

19. Classify accelerators.

20. What are the parameters of linear accelerators.

21. When does a charged particle move in a spiral in a magnetic field? What determines the pitch of the helix? Support your answers with formulas.

22. What are charged particle accelerators? What are they and what are they characterized by?

23. Why are cyclotrons not used to accelerate electrons?

24. What is the principle of autophasing? Where is it used?

25. When is the EMF of self-induction greater - when closing or opening the DC circuit?

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

27. What are the phenomena of self-induction and mutual induction? Calculate the induction emf for both cases.

28. What is the physical meaning of relaxation time? Prove that it has the dimension of time.

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

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

31. Give the relationship between the points in the primary and secondary windings of a step-up transformer.

Table 5.1 shows the comparative characteristics of the electric and magnetic fields.

Table 5.1

The magnetic field has energy. To verify this, consider an electrical circuit containing a solenoid having inductance and resistance (Fig. 6.6). When the key is opened TO the current does not immediately drop to zero. For some time, it continues to flow, supported by the electromotive force of self-induction arising in the coil, and at the same time, heat is released on the resistance, according to the Joule-Lenz law. The question arises due to what energy reserves heat is released, because the circuit is open and the external source is turned off.

When the current in the circuit decreases, the magnetic field induction also decreases. Therefore, it is possible, apparently, to speak of the energy of the electric current or the energy of the magnetic field created by the current. In the case of direct currents, it is impossible to unambiguously determine where this energy is localized. The answer to this question can be given by studying alternating magnetic fields or electromagnetic waves. In electromagnetic waves, alternating magnetic fields can exist without currents to support them. Since electromagnetic waves carry energy, it can be concluded that the energy is concentrated in a magnetic field.

Let us find the value of the energy of the magnetic field. It follows from the law of conservation of energy that when the current stops, the magnetic field will disappear, and all the energy of the magnetic field will turn into thermal energy. According to the Joule-Lenz law, for a short time on the resistance R the amount of heat released. According to Ohm's law, the current I equals

Given this equality, the released amount of heat can be written as:

in this expression, since the current decreases, and the released heat . The dependence of the magnetic flux on the current strength can be represented graphically (Fig. 6.7). It is obvious that the amount of heat released over time is equal to the initial supply of magnetic energy and is determined by the area of ​​the triangle made up of a straight line, a straight line and an axis. This area is equal to . Thus, the energy of the magnetic field created by the current I in an inductor L, is equal to

Compare the expression for the magnetic energy stored in an inductor with the expression for the electric field energy stored in a capacitor:

The energy of the electric field in the capacitor is proportional to the square of the charge, the energy of the magnetic field stored in the inductor is proportional to the square of the current strength, that is, it depends on the speed of the charges. Recall that a magnetic field is created by moving charges.

The work of the induction current is accompanied by heating of the conductor due to the energy of the magnetic field, which cannot disappear without a trace. The solenoid thus serves as a kind of energy reservoir, the value of which is calculated by the formula

Since the magnetic field inside the solenoid is uniform, the energy density of the magnetic field stored in the solenoid is equal to the energy divided by the volume of the solenoid:

Example

Let us determine the energy of the magnetic field of the solenoid. Conventional laboratory solenoid length 10 cm, cross-sectional area 75 cm 2 and the number of turns wound in several layers, equal to 3400, has inductance. The resistance of such a solenoid is 50 Ohm. When using a 6-volt battery, a current will be established. The magnetic energy stored in the solenoid is equal to This is a small energy. However, this energy is proportional to the square of the current strength and can reach large values. So, for example, in electromagnets used for research, the magnetic induction at maximum current is usually from 1 to 1.5 Tl. The magnetic permeability of iron reaches values ​​of hundreds and thousands of units, therefore, in an electromagnet, most of the energy is concentrated in the gap between the poles of the electromagnet. If the gap volume is 0.2, then the stored energy

/ = 1.8 J.

That's already a lot of energy! If, without special precautions, the electromagnet circuit is quickly opened, then at the instantaneous power will be P = 1,8 MW.

Mutual induction

Similarly, if a current of strength flows in circuit 2, it creates a magnetic flux through circuit 1:

The coefficients of proportionality and are called the mutual inductance of the circuits. From (6.6) and (6.7) it can be seen that the mutual inductance is numerically equal to the magnetic flux through one of the circuits at a unit current in the other circuit. The coefficients and depend on the shape, size, mutual arrangement of the contours, as well as on the magnetic properties of the medium surrounding the contours.

It can be shown that in the absence of ferromagnets the coefficients and are the same: . This property is called the reciprocity theorem. The reciprocity theorem allows us not to make a distinction between and , but simply talk about the mutual inductance of two circuits. According to the reciprocity theorem, if the same currents flow in the circuits, then the magnetic flux through circuit 1, created by the current in circuit 2, is equal to the magnetic flux through circuit 2, created by the current in circuit 1.

If the circuits are stationary and there are no ferromagnets near them, then when the current strength changes in one of the circuits, an electromotive force of induction arises in the other circuit. This phenomenon is called the phenomenon of mutual induction. According to the law of electromagnetic induction, the electromotive forces of induction arising in circuits 1 and 2 are equal, respectively

For a given direction, the current will depend on the choice of a positive normal to the surface bounded by contour 2. Positive directions for currents (and electromotive forces) in both contours can be chosen arbitrarily. For a given current direction, the direction of the positive normal to the contour surface is determined by the rule of the right screw. If these directions are chosen, the value must be considered positive when, at positive currents, the magnetic fluxes of mutual induction through the circuits also turn out to be positive, that is, they coincide in sign with the self-induction fluxes.

In other words, if with positive currents in both circuits they "magnetize" each other, otherwise. In particular cases, it is possible to set in advance the positive directions for bypassing the contours in order to obtain the desired sign of the quantity .

Example

Without a strong cellular signal, your phone becomes more susceptible to electromagnetic interference. This happens due to a change in the signal due to the phenomenon of mutual induction. An example of such an effect is the deterioration of phone reception when approaching a TV or radio.

6.6. Examples on the application of the phenomenon
electromagnetic induction

Self-induction. Mutual induction.

An electric current flowing in a closed circuit creates a magnetic field around itself, the induction of which, according to the Biot-Savart-Laplace law, is proportional to the current B~I :

Magnetic flux is proportional to magnetic induction F~V .

Then the magnetic flux coupled to the circuit F proportional to the current in the circuit:

Proportionality factor L in this formula is called self-induction coefficient or inductance circuit (coil). The SI unit of inductance is called Henry (GN). The inductance of a circuit or coil is 1 H if, at a DC current of 1 A, the own flux is 1 Wb:

When the current strength in the circuit changes, the magnetic flux coupled to it will also change. The occurrence of an EMF of induction in a conducting circuit with a change in the current strength in it is called self-induction.

self induction is an important special case of electromagnetic induction, when a changing magnetic flux, causing an EMF of induction, is created by a current in the circuit itself. If the current in the circuit under consideration changes for some reason, then the magnetic field of this current changes, and, consequently, the own magnetic flux penetrating the circuit. In the circuit, an EMF of self-induction occurs, which, according to the Lenz rule, prevents a change in the current in the circuit.

EMF self-induction, arising in the circuit, according to Faraday's law is equal to:

.

If the contour is not deformed and the magnetic permeability of the medium does not change, then L= const And

where the “−” sign, due to the Lenz rule, shows that the presence of inductance in the circuit leads to slow change current in it.

The EMF of self-induction is directly proportional to the inductance of the coil and the rate of change of the current strength in it.

The second special case of electromagnetic induction: mutual induction − the phenomenon of the occurrence of EMF in one of the circuits when the current strength changes in the other circuit.

If the current I 1 changes, then EMF is induced in circuit 2, which, according to Faraday's law, is equal and opposite in sign to the rate of change of the magnetic flux Ф 12 created by the current in circuit 1 and penetrating the 2nd:

,

Where L 12– coefficient of proportionality, depending on the geometry of circuits 1 and 2 and their relative position.

If the current I 2 changes, then EMF is induced in circuit 1, which, according to Faraday's law, is equal and opposite in sign to the rate of change of the magnetic flux Ф 21 created by the current in the 2nd circuit and penetrating the 1st:

.

Proportionality coefficients L 12 And L 21 called the mutual inductance of the circuits. Experimental calculations show that L 12 = L 21.

The magnetic field has energy. Just as a charged capacitor has a supply of electrical energy, a coil with current flowing through its coils has a supply of magnetic energy. If you turn on an electric lamp in parallel with a coil with a large inductance in a DC electric circuit, then when the key is opened, a short flash of the lamp is observed (Fig. 1). The current in the circuit arises under the action of self-induction EMF. The source of energy released in this case in the electrical circuit is the magnetic field of the coil.

Fig.1. The magnetic energy of the coil.

When the key K is opened, the lamp flashes brightly

formula for energy W m of the magnetic field can be obtained by a graphical method, depicting on the graph the dependence of the magnetic flux Φ ( I) from current I(Fig. 2). The total amount of released heat, equal to the initial energy reserve of the magnetic field, is determined by the area shown in Fig. 2 triangles.