Magnetic induction of a circular coil. Determination of magnetic field induction on the axis of circular current. Magnetic moment of a coil with current. Vortex nature of the magnetic field

Magnetic field of current:

A magnetic field created around electric charges as they move. Since the movement of electric charges represents an electric current, around any conductor with current there is always current magnetic field.

To verify the existence of a magnetic field of current, let’s bring an ordinary compass from above to the conductor through which electric current flows. The compass needle will immediately deviate to the side. We bring the compass to the conductor with current from below - the compass needle will deviate in the other direction (Figure 1).

Let us apply the Biot–Savart–Laplace law to calculate the magnetic fields of the simplest currents. Let's consider the magnetic field of direct current.

All vectors dB from arbitrary elementary sections dl have the same direction. Therefore, addition of vectors can be replaced by addition of modules.

Let the point at which the magnetic field is determined be located at a distance b from the wire. From the figure it can be seen that:

;

Substituting the found values r and d l into the Biot-Savart-Laplace law, we get:

For final conductor angle α varies from , to. Then

For infinitely long conductor , and , then

or, which is more convenient for calculations, .

Direct current magnetic induction lines are a system of concentric circles enclosing the current.

21. Biot-Savart-Laplace law and its application to the calculation of the magnetic field induction of a circular current.

Magnetic field of a circular conductor carrying current.

22. Magnetic moment of a coil with current. Vortex nature of the magnetic field.

The magnetic moment of a coil with current is a physical quantity, like any other magnetic moment, that characterizes the magnetic properties of a given system. In our case, the system is represented by a circular coil with current. This current creates a magnetic field that interacts with the external magnetic field. This can be either the field of the earth or the field of a permanent or electromagnet.

Figure - 1 circular turn with current

A circular coil with current can be represented as a short magnet. Moreover, this magnet will be directed perpendicular to the plane of the coil. The location of the poles of such a magnet is determined using the gimlet rule. According to which the north plus will be located behind the plane of the coil if the current in it moves clockwise.

Figure-2 Imaginary strip magnet on the coil axis

This magnet, that is, our circular coil with current, like any other magnet, will be affected by an external magnetic field. If this field is uniform, then a torque will arise that will tend to turn the coil. The field will rotate the coil so that its axis is located along the field. In this case, the field lines of the coil itself, like a small magnet, must coincide in direction with the external field.

If the external field is not uniform, then translational motion will be added to the torque. This movement will occur due to the fact that sections of the field with higher induction will attract our magnet in the form of a coil more than areas with lower induction. And the coil will begin to move towards the field with greater induction.

The magnitude of the magnetic moment of a circular coil with current can be determined by the formula.

Where, I is the current flowing through the turn

S area of ​​the turn with current

n normal to the plane in which the coil is located

Thus, from the formula it is clear that the magnetic moment of a coil is a vector quantity. That is, in addition to the magnitude of the force, that is, its modulus, it also has a direction. The magnetic moment received this property due to the fact that it includes the normal vector to the plane of the coil.

All elements (dl) of the circular current create induction (dB) in the center of the circle;

from (61)

(62)

Ampere's law sets the force acting on a current-carrying conductor (force modulus) in a magnetic field:

Ampere force direction determined using the left hand rule.

Interaction of two conductors. Let us consider the interaction of two infinite rectilinear parallel conductors with currents and located at a distance R.

Using Ampere's law (63) and the formula for magnetic induction (60), taking into account that for the force of interaction of two currents we obtain

(64)

Lorentz force– force acting on a charge moving in a magnetic field:

(65) or (66)

The direction of the force is determined using the left-hand rule (on a positive charge).

We find the radius of rotation r from the equality

(67)

Treatment period:

(68), from here (69) i.e. the period of particle motion does not depend on their speed. This is used in particle accelerators – cyclotrons.

Accelerators are divided into: linear, cyclic and induction. To accelerate relativistic particles, they use: phasotron - the frequency of the alternating electric field increases, synchrotron - the magnetic field increases, synchrotron - the frequency and magnetic field increase.

Magnetic induction vector flux(magnetic flux) through the area dS is called scalar physical quantity equal to

(70)

(71) where is the projection of the vector onto the normal direction ,

α – angle between and

Total flow value:

. (72)

Let us consider as an example the magnetic field of an infinite rectilinear conductor with current I located in a vacuum. Vector circulation along an arbitrary line of magnetic induction - a circle of radius r:
Because at all points of the induction line is equal in modulus and is directed tangentially to the line, so , hence:
Those. The circulation of the magnetic induction vector in a vacuum is the same along all lines of magnetic induction and is equal to the product of the magnetic constant and the current strength. This conclusion is valid for any arbitrary closed circuit if current flows inside it. If the circuit does not cover the current, then the vector circulation along this circuit is equal to 0. If there are many currents, then the algebraic sum of the currents is taken.

Theorem: The circulation of magnetic field induction in a vacuum along an arbitrary closed circuit L is equal to the product of the magnetic constant and the algebraic sum of the currents covered by this circuit. This law can also be written:

(73)

Lecture 9

3.2.(2 hours) Magnetic properties of matter. Molecular currents. Dia -, para - and ferromagnets. Magnetization vector. Magnetic susceptibility and magnetic permeability. Introduction to nuclear magnetic resonance and electron paramagnetic resonance.

Magnetic moments of electrons and atoms. All substances placed in a magnetic field become magnetized. From the point of view of the structure of atoms, an electron moving in a circular orbit has orbital magnetic moment:

(74) its modulus

(75) where - current strength,

Rotation frequency,

S– orbital area.

The direction of the vector is determined by the gimlet rule. An electron moving in orbit also has a mechanical angular momentum, the magnitude of which is

- orbital mechanical moment of the electron. (76) where ,

.

Directions and opposite, because the charge of the electron is negative. From (75) and (76) we obtain

(77) where - gyromagnetic ratio. (78)

The formula is also valid for non-circular orbits. The value of g was determined experimentally by Einstein and de Haas (1915). It turned out to be equal to , that is, twice as large as (78). Then it was assumed, and subsequently proven, that in addition to orbital angular momentum, the electron has its own mechanical angular momentum, called spin. The electron spin corresponds to its own (spin) magnetic moment: . The quantity is called the gyromagnetic ratio of spin moments. The projection of the intrinsic magnetic moment onto the direction of the vector can take only one of the following two values ±еħ/2m= , where ħ= , h is Planck’s constant, is the Bohr magneton, which is a unit of the magnetic moment of an electron. The total magnetic moment of an atom (molecule) is equal to the vector sum of the magnetic moments of (orbital and spin) electrons: .

Dia – and paramagnetism. Every substance is magnetic, i.e. it is capable of acquiring a magnetic moment under the influence of a magnetic field, i.e. magnetize.

If the electron’s orbit is oriented relative to the external field vector in an arbitrary way, making ےα with it, then the orbit and the vector will go into rotation, which is called precession(movement of the top). Precessional motion is equivalent to current. The induced components of the magnetic fields of atoms add up and form the substance’s own magnetic field, which is superimposed on the external magnetic field and a resulting magnetic field is formed inside the magnet.

Diamagnets– these are substances in which the magnetic field decreases. For them, the magnetic permeability slightly less than 1 is μ ≈ 0.999935. (Explained by the action of Lenz's rule). Diamagnetism is characteristic of all substances.

Paramagnets– substances in which the magnetic field increases under the influence of an external field, for them μ is greater than 1, for example, μ ≈ 1.00047. Paramagnetic elements include rare earth elements: Pt, Al, CuSO 4, etc. Explained by the orientation of the orbital and spin magnetic moments of atoms in a magnetic field. When the external magnetic field ceases, the orientation is destroyed by the thermal movement of atoms and the paramagnet is demagnetized. The magnetic permeability of paramagnetic materials exceeds that of diamagnetic materials.

To quantitatively describe the magnetization of magnets, a vector quantity is introduced - magnetization, determined by the magnetic moment per unit volume of the magnet:

(79) where - the magnetic moment of a magnet, which is the vector sum of the magnetic moments of individual molecules. The vector of the resulting magnetic field in the magnet is equal to the vector sum of the magnetic inductions of the external field and the field of microcurrents (molecular currents): , from here In weak fields, magnetization is proportional to the strength of the field causing magnetization, i.e. , where χ – magnetic susceptibility of a substance. For diamagnetic materials it is negative, for paramagnetic materials it is positive. From the above formulas: Here , using this formula we arrive at the well-known formula

Phenomenon electron paramagnetic resonance was discovered in Kazan in 1945 by scientist E.K. Zavoisky, an employee of Kazan University. The essence of the phenomenon lies in the resonant absorption of a high-frequency electromagnetic field when it acts on a paramagnetic substance that is in a constant magnetic field. In this case, the frequency of the Larmor procession of electron spins coincides with the frequency of the external electromagnetic field and the electron absorbs this energy.

The magnetic moments of atomic nuclei are much weaker than the magnetic moments of electrons, so nuclear magnetic resonance was discovered later than electron magnetic resonance, in 1949 in the USA. The process is similar to the electronic one, but has become more widely used for the study of substances. The pinnacle of this application is the creation of NMR tomographs.

Ferromagnets. These include: iron, cobalt, nickel, gadolinium, their alloys and compounds. μ>>1 is several thousand.

I us – magnetic saturation.

Upon saturation, an increasing number of magnetic moments are oriented.

A characteristic feature of ferromagnets is that for them the dependence of I on H (and therefore B on H) has the form of a loop, which is called a hysteresis loop: 0 – demagnetized; 1 – saturation (); 2 – residual magnetization (), permanent magnets; 3 – demagnetization ( – coercive force); then it repeats.

Ferromagnets with low coercive force are called 1) soft, and with high coercive force - 2) hard. The former are used for the cores of transformers and electrical machines (motors and generators), the latter - for permanent magnets. Curie point– the temperature at which a ferromagnetic material loses its magnetic properties and turns into a paramagnetic material. The process of magnetization of ferromagnets is accompanied by a change in their linear dimensions and volume. This phenomenon is called magnetostriction. Ferromagnets have a domain structure: microscopic volumes in which the magnetic moments are oriented in the same way. In a non-magnetized state, the magnetic moments of the domains are directed randomly and the resulting field is zero. When a ferromagnet is magnetized, the magnetic moments of the domains rotate abruptly and are established along the field and the ferromagnet is magnetized. As soon as all domains are oriented, the magnetization reaches saturation. With residual magnetization () – some of the domains are oriented.

There are antiferromagnets (compounds MnO, MnF 2, FeO, FeCl 2).

Recently, they have gained great importance ferrites– semiconductor ferromagnets, chemical compounds such as , where Me is a divalent metal ion (Mn, Co, Ni, Cu, Zn, Cd, Fe). They have noticeable ferromagnetic properties and high electrical resistivity (millions of times greater than that of metals). They are widely used in electrical engineering and radio engineering.

Goal of the work : study the properties of the magnetic field, become familiar with the concept of magnetic induction. Determine the magnetic field induction on the axis of the circular current.

Theoretical introduction. A magnetic field. The existence of a magnetic field in nature is manifested in numerous phenomena, the simplest of which are the interaction of moving charges (currents), current and a permanent magnet, two permanent magnets. A magnetic field vector . This means that for its quantitative description at each point in space it is necessary to set the magnetic induction vector. Sometimes this quantity is simply called magnetic induction . The direction of the magnetic induction vector coincides with the direction of the magnetic needle located at the point in space under consideration and free from other influences.

Since the magnetic field is a force field, it is depicted using magnetic induction lines – lines, the tangents to which at each point coincide with the direction of the magnetic induction vector at these points of the field. It is customary to draw through a single area perpendicular to , a number of magnetic induction lines equal to the magnitude of the magnetic induction. Thus, the density of the lines corresponds to the value IN . Experiments show that there are no magnetic charges in nature. The consequence of this is that the magnetic induction lines are closed. The magnetic field is called homogeneous, if the induction vectors at all points of this field are the same, that is, equal in magnitude and have the same directions.

For the magnetic field it is true superposition principle: the magnetic induction of the resulting field created by several currents or moving charges is equal to vector sum magnetic induction fields created by each current or moving charge.

In a uniform magnetic field, a straight conductor is acted upon by Ampere power:

where is a vector equal in magnitude to the length of the conductor l and coinciding with the direction of the current I in this guide.

The direction of the Ampere force is determined right screw rule(vectors , and form a right-handed screw system): if a screw with a right-hand thread is placed perpendicular to the plane formed by the vectors and , and rotated from to at the smallest angle, then the translational movement of the screw will indicate the direction of the force. In scalar form, relation (1) can be written as follows way:

F = I× l× B× sin a or 2).

From the last relation it follows physical meaning of magnetic induction : magnetic induction of a uniform field is numerically equal to the force acting on a conductor with a current of 1 A, 1 m long, located perpendicular to the direction of the field.

The SI unit of magnetic induction is Tesla (T): .

Magnetic field of circular current. Electric current not only interacts with a magnetic field, but also creates it. Experience shows that in a vacuum a current element creates a magnetic field with induction at a point in space

(3) ,

where is the proportionality coefficient, m 0 =4p×10 -7 H/m– magnetic constant, – vector numerically equal to the length of the conductor element and coinciding in direction with the elementary current, – radius vector drawn from the conductor element to the field point under consideration, r – modulus of the radius vector. Relationship (3) was established experimentally by Biot and Savart, analyzed by Laplace and is therefore called Biot-Savart-Laplace law. According to the rule of the right screw, the magnetic induction vector at the point under consideration turns out to be perpendicular to the current element and the radius vector.

Based on the Biot-Savart-Laplace law and the principle of superposition, the magnetic fields of electric currents flowing in conductors of arbitrary configuration are calculated by integrating over the entire length of the conductor. For example, the magnetic induction of a magnetic field at the center of a circular coil with a radius R , through which current flows I , is equal to:

The magnetic induction lines of circular and forward currents are shown in Figure 1. On the axis of the circular current, the magnetic induction line is straight. The direction of magnetic induction is related to the direction of current in the circuit right screw rule. When applied to circular current, it can be formulated as follows: if a screw with a right-hand thread is rotated in the direction of the circular current, then the translational movement of the screw will indicate the direction of the magnetic induction lines, the tangents to which at each point coincide with the magnetic induction vector.

All elements of a circular conductor with current create magnetic fields in the center of the same direction - along the normal from the turn. therefore, all elements of the coil are perpendicular to the radius vector, then ; since the distances from all elements of the conductor to the center of the turn are the same and equal to the radius of the turn. That's why:

Direct conductor field.

As the integration constant, we choose the angle α (the angle between the vectors dB And r ), and express all other quantities through it. From the figure it follows that:

Let's substitute these expressions into the formula of the Biot-Savart-Laplace law:

And - the angles at which the ends of the conductor are visible from the point at which the magnetic induction is measured. Let's substitute it into the formula:

In the case of an infinitely long conductor ( and ) we have:

Application of Ampere's law.

Interaction of parallel currents

Consider two infinite rectilinear parallel currents directed in one direction I 1 And I 2, the distance between which is R. Each of the conductors creates a magnetic field, which acts according to Ampere's law on the other conductor with current. Current I 1 creates a magnetic field around itself, the lines of magnetic induction of which are concentric circles. Vector direction IN , is determined by the rule of the right screw, its module is equal to:

Direction of force d F 1 , with which the field B 1 acts on the area dl the second current is determined by the left-hand rule. The force modulus taking into account the fact that the angle α between the current elements I 2 and vector B 1 straight, equal

Substituting the value B 1 . we get:

By similar reasoning, one can prove that

It follows that, that is, two parallel currents are attracted to each other with the same force. If the currents are in the opposite direction, then using the left-hand rule, it can be shown that there is a repulsive force between them.

Interaction force per unit length:

Behavior of a current-carrying circuit in a magnetic field.

Let us introduce a square frame with side l with current I into the magnetic field B, the rotational moment of a pair of Ampere forces will act on the circuit:

Magnetic moment of the circuit,

Magnetic induction at the field point where the circuit is located

The current-carrying circuit tends to establish itself in a magnetic field so that the flux through it is maximum and the torque is minimum.

Magnetic induction at a given point in the field is numerically equal to the maximum torque acting at a given point in the field on a circuit with a unit magnetic moment.

Law of total current.

Let us find the circulation of vector B along a closed contour. Let's take a long conductor with current I as the field source, and a field line of radius r as the contour.

Let us extend this conclusion to a circuit of any shape, covering any number of currents. Total current law:

The circulation of the magnetic induction vector along a closed circuit is proportional to the algebraic sum of the currents covered by this circuit.

Application of the total current law to calculate fields

Field inside an infinitely long solenoid:

where τ is the linear density of winding turns, l S– solenoid length, N– number of turns.

Let the closed contour be a rectangle of length X, which braids the turns, then induction IN along this circuit:

Let's find the inductance of this solenoid:

Toroid field(wire wound around a frame in the form of a torus).

R– average radius of the torus, N– number of turns, where – linear density of winding turns.

Let's take a line of force with radius R as a contour.

Hall effect

Consider a metal plate placed in a magnetic field. An electric current is passed through the plate. A potential difference arises. Since the magnetic field affects moving electric charges (electrons), they will be subject to the Lorentz force, moving electrons to the upper edge of the plate, and, therefore, an excess of positive charge will form at the lower edge of the plate. Thus, a potential difference is created between the upper and lower edges. The process of moving electrons will continue until the force acting from the electric field is balanced by the Lorentz force.

Where d– plate length, A– plate width, – Hall potential difference.

Law of electromagnetic induction.

Magnetic flux

where α is the angle between IN and outer perpendicular to the contour area.

For any change in magnetic flux over time. Thus, the induced emf occurs both when the area of ​​the circuit changes and when the angle α changes. Induction emf is the first derivative of magnetic flux with respect to time:

If the circuit is closed, then an electric current begins to flow through it, called an induction current:

Where R– circuit resistance. The current arises due to a change in magnetic flux.

Lenz's rule.

An induced current always has a direction such that the magnetic flux created by this current prevents the change in the magnetic flux that caused this current. The current has such a direction as to interfere with the cause that caused it.

Rotation of the frame in a magnetic field.

Let us assume that the frame rotates in a magnetic field with an angular velocity ω, so that the angle α is equal to . in this case the magnetic flux is:

Consequently, a frame rotating in a magnetic field is a source of alternating current.

Eddy currents (Foucault currents).

Eddy currents or Foucault currents arise in the thickness of conductors that are in an alternating magnetic field, creating an alternating magnetic flux. Foucault currents lead to heating of conductors and, consequently, to electrical losses.

The phenomenon of self-induction.

With any change in magnetic flux, an induced emf occurs. Let us assume that there is an inductor through which electric current flows. According to the formula, in this case a magnetic flux is created in the coil. With any change in current in the coil, the magnetic flux changes and, therefore, an emf occurs, called self-induction emf ():

Maxwell's system of equations.

The electric field is a set of mutually related and mutually changing magnetic fields. Maxwell established a quantitative relationship between the quantities characterizing electric and magnetic fields.

Maxwell's first equation.

From Faraday's law of electromagnetic induction it follows that with any change in the magnetic flux, an emf appears. Maxwell suggested that the appearance of EMF in the surrounding space is associated with the appearance in the surrounding space vortex electromagnetic field. The conducting circuit plays the role of a device that detects the appearance of this electric field in the surrounding space.

The physical meaning of Maxwell's first equation: any change in time of the magnetic field leads to the appearance of a vortex electric field in the surrounding space.

Maxwell's second equation. Bias current.

The capacitor is connected to the DC circuit. Suppose that a circuit containing a capacitor is connected to a constant voltage source. The capacitor charges and the current in the circuit stops. If a capacitor is connected to an alternating voltage circuit, the current in the circuit does not stop. This is due to the process of continuous recharging of the capacitor, as a result of which a time-varying electric field appears between the plates of the capacitor. Maxwell suggested that a displacement current arises in the space between the plates of the capacitor, the density of which is determined by the rate of change of the electric field over time. Of all the properties inherent in electric current, Maxwell attributed one single property to displacement current: the ability to create a magnetic field in the surrounding space. Maxwell suggested that conduction current lines on the capacitor plates do not stop, but continuously transform into displacement current lines. Thus:

Thus, the current density is:

where is the conduction current density, is the displacement current density.

According to the law of total current:

The physical meaning of Maxwell's second equation: the source of the magnetic field is both conduction currents and a time-varying electric field.

Maxwell's third equation (Gauss's theorem).

The flux of the electrostatic field strength vector through a closed surface is equal to the charge contained inside this surface:

Physical meaning of Maxwell's fourth equation: lines electrostatic fields begin and end on free electric charges. That is, the source of the electrostatic field is electric charges.

Maxwell's fourth equation (magnetic flux continuity principle)

The physical meaning of Maxwell's fourth equation: the lines of the magnetic induction vector do not begin or end anywhere, they are continuous and closed on themselves.

Magnetic properties of substances.

Magnetic field strength.

The main characteristic of a magnetic field is the magnetic induction vector, which determines the force effect of the magnetic field on moving charges and currents; the magnetic induction vector depends on the properties of the medium where the magnetic field is created. Therefore, a characteristic is introduced that depends only on the currents associated with the field, but does not depend on the properties of the medium where the field exists. This characteristic is called magnetic field strength and is denoted by the letter H.

If a magnetic field in a vacuum is considered, then the intensity

where is the magnetic constant of vacuum. Unit of tension Ampere/meter.

Magnetic field in matter.

If the entire space surrounding the currents is filled with a homogeneous substance, then the magnetic field induction will change, but the distributed field will not change, that is, the magnetic field induction in the substance is proportional to the magnetic induction in vacuum. - magnetic permeability of the medium. Magnetic permeability shows how many times the magnetic field in a substance differs from the magnetic field in a vacuum. The value can be either less or greater than one, that is, the magnetic field in a substance can be either less or greater than the magnetic field in a vacuum.

Magnetization vector. Every substance is magnetic, that is, it is capable of acquiring a magnetic moment under the influence of an external magnetic field - being magnetized. The electrons of atoms under the influence of a mutual magnetic field undergo precessional motion - a movement in which the angle between the magnetic moment and the direction of the magnetic field remains constant. In this case, the magnetic moment rotates around the magnetic field with a constant angular velocity ω. Precessional motion is equivalent to circular current. Since the microcurrent is induced by an external magnetic field, then, according to Lenz’s rule, the atom has a magnetic field component directed opposite to the external field. The induced component of magnetic fields adds up and forms its own magnetic field in the substance, directed opposite to the external magnetic field, and, therefore, weakening this field. This effect is called the diamagnetic effect, and substances in which the diamagnetic effect occurs are called diamagnetic substances or diamagnetic substances. In the absence of an external magnetic field, a diamagnetic material is nonmagnetic, since the magnetic moments of the electrons are mutually compensated and the total magnetic moment of the atom is zero. Since the diamagnetic effect is caused by the action of an external magnetic field on the electrons of the atoms of a substance, diamagnetism is characteristic of ALL SUBSTANCES.

Paramagnetic substances are substances in which, even in the absence of an external magnetic field, atoms and molecules have their own magnetic moment. However, in the absence of an external magnetic field, the magnetic moments of different atoms and molecules are randomly oriented. In this case, the magnetic moment of any macroscopic volume of matter is zero. When a paramagnetic substance is introduced into an external magnetic field, the magnetic moments are oriented in the direction of the external magnetic field, and a magnetic moment appears directed along the direction of the magnetic field. However, the total magnetic field arising in a paramagnetic substance significantly overlaps the diamagnetic effect.

The magnetization of a substance is the magnetic moment per unit volume of the substance.

where is the magnetic moment of the entire magnet, equal to the vector sum of the magnetic moments of individual atoms and molecules.

The magnetic field in a substance consists of two fields: an external field and a field created by the magnetized substance:

(reads "hee") is the magnetic susceptibility of the substance.

Let's substitute formulas (2), (3), (4) into formula (1):

The coefficient is a dimensionless quantity.

For diamagnetic materials (this means that the field of molecular currents is opposite to the external field).

For paramagnetic materials (this means that the field of molecular currents coincides with the external field).

Therefore, for diamagnetic materials, and for paramagnetic materials. And N .

Hysteresis loop.

Magnetization dependence J on the strength of the external magnetic field H forms a so-called “hysteresis loop”. At the beginning (section 0-1) the ferromagnet is magnetized, and the magnetization does not occur linearly, and at point 1 saturation is achieved, that is, with a further increase in the magnetic field strength, the current growth stops. If you start to increase the strength of the magnetizing field, then the decrease in magnetization follows the curve 1-2 , lying above the curve 0-1 . When residual magnetization is observed (). The existence of permanent magnets is associated with the presence of residual magnetization. The magnetization goes to zero at point 3, at a negative value of the magnetic field, which is called the coercive force. With a further increase in the opposite field, the ferromagnet is remagnetized (curve 3-4). Then the ferromagnet can be demagnetized again (curve 4-5-6) and magnetize again until saturation (curve 6-1). Ferromagnets with low coercivity (with small values ​​of ) are called soft ferromagnets, and they correspond to a narrow hysteresis loop. Ferromagnets with a high coercive force are called hard ferromagnets. For each ferromagnet there is a certain temperature, called the Curie point, at which the ferromagnet loses its ferromagnetic properties.

The nature of ferromagnetism.

According to Weiss's ideas. Ferromagnets at temperatures below the Curie point have a domain structure, namely, ferromagnets consist of macroscopic regions called domains, each of which has its own magnetic moment, which is the sum of the magnetic moments of a large number of atoms of a substance oriented in the same direction. In the absence of an external magnetic field, the domains are randomly oriented and the resulting magnetic moment of the ferromagnet is generally zero. When an external magnetic field is applied, the magnetic moments of the domains begin to be oriented in the direction of the field. In this case, the magnetization of the substance increases. At a certain value of the external magnetic field strength, all domains are oriented along the field direction. In this case, the growth of magnetization stops. When the external magnetic field strength decreases, the magnetization begins to decrease again; however, not all domains are misoriented at the same time, so the decrease in magnetization occurs more slowly, and when the magnetic field strength is equal to zero, a fairly strong orienting connection remains between some domains, which leads to the presence of residual magnetization coinciding with direction of the previously existing magnetic field.

To break this connection, it is necessary to apply a magnetic field in the opposite direction. At temperatures above the Curie point, the intensity of thermal motion increases. Chaotic thermal movement breaks the bonds within the domains, that is, the preferential orientation of the domains themselves is lost. Thus, the ferromagnet loses its ferromagnetic properties.

Exam questions:

1) Electric charge. Law of conservation of electric charge. Coulomb's law.

2) Electric field strength. The physical meaning of tension. Field strength of a point charge. Electric field lines.

3) Two definitions of potentials. Work on moving a charge in an electric field. The connection between tension and potential. Work along a closed trajectory. Circulation theorem.

4) Electrical capacity. Capacitors. Series and parallel connection of capacitors. Capacitance of a parallel plate capacitor.

5) Electric current. Conditions for the existence of electric current. Current strength, current density. Units of current measurement.

6) Ohm's law for a homogeneous section of the chain. Electrical resistance. Dependence of resistance on the cross-sectional length of the conductor material. Dependence of resistance on temperature. Serial and parallel connection of conductors.

7) Outside forces. EMF. Potential difference and voltage. Ohm's law for a non-uniform section of a circuit. Ohm's law for a closed circuit.

8) Heating of conductors with electric current. Joule-Lenz law. Electric current power.

9) Magnetic field. Ampere power. Left hand rule.

10) Movement of a charged particle in a magnetic field. Lorentz force.

11) Magnetic flux. Faraday's law of electromagnetic induction. Lenz's rule. The phenomenon of self-induction. Self-induced emf.

Let a direct electric current of force I flow along a flat circular contour of radius R. Let us find the field induction in the center of the ring at point O
Let us mentally divide the ring into small sections that can be considered rectilinear, and apply the Biot-Savarre-Laplace law to determine the induction of the field created by this element in the center of the ring. In this case, the vector of the current element (IΔl)k and the vector rk connecting this element with the observation point (the center of the ring) are perpendicular, therefore sinα = 1. The induction vector of the field created by the selected section of the ring is directed along the axis of the ring, and its modulus is equal to

For any other element of the ring, the situation is absolutely similar - the induction vector is also directed along the axis of the ring, and its module is determined by formula (1). Therefore, the summation of these vectors is carried out elementary and is reduced to the summation of the lengths of the sections of the ring

Let's complicate the problem - find the field induction at point A, located on the axis of the ring at a distance z from its center.
As before, we select a small section of the ring (IΔl)k and construct the induction vector of the field ΔBk created by this element at the point in question. This vector is perpendicular to the vector r connecting the selected area with the observation point. Vectors (IΔl)k and rk, as before, are perpendicular, so sinα = 1. Since the ring has axial symmetry, the total field induction vector at point A must be directed along the axis of the ring. The same conclusion about the direction of the total induction vector can be reached if we notice that each selected section of the ring has a symmetrical one on the opposite side, and the sum of two symmetrical vectors is directed along the axis of the ring. Thus, in order to determine the module of the total induction vector, it is necessary to sum up the projections of the vectors onto the axis of the ring. This operation is not particularly difficult, given that the distances from all points of the ring to the observation point are the same rk = √(R2+ z2), and the angles φ between the vectors ΔBk and the axis of the ring are the same. Let us write down the expression for the modulus of the desired total induction vector

From the figure it follows that cosφ = R/r, taking into account the expression for the distance r, we obtain the final expression for the field induction vector

As one would expect, in the center of the ring (at z = 0) formula (3) transforms into the previously obtained formula (2).

Using the general method discussed here, it is possible to calculate the field induction at an arbitrary point. The system under consideration has axial symmetry, so it is enough to find the field distribution in a plane perpendicular to the plane of the ring and passing through its center. Let the ring lie in the xOy plane (Fig. 433), and the field is calculated in the yOz plane. The ring should be divided into small sections visible from the center at an angle Δφ and the fields created by these sections should be summed up. It can be shown (try it yourself) that the components of the magnetic induction vector of the field created by one selected current element at a point with coordinates (y, z) are calculated using the formulas:

Let us consider the expression for the field induction on the ring axis at distances significantly larger than the ring radius z >> R. In this case, formula (3) is simplified and takes the form

Where IπR2 = IS = pm is the product of the current strength and the area of ​​the circuit, that is, the magnetic moment of the ring. This formula coincides (if, as usual, replace μo in the numerator with εo in the denominator) with the expression for the electric field strength of a dipole on its axis.
This coincidence is not accidental; moreover, it can be shown that such a correspondence is valid for any point in the field located at large distances from the ring. In fact, a small circuit with current is a magnetic dipole (two identical small oppositely directed current elements) - therefore its field coincides with the field of an electric dipole. To more clearly emphasize this fact, a picture of the magnetic field lines of the ring at large distances from it is shown (compare with a similar picture for the field of an electric dipole).