Magnetic field of a multilayer solenoid online. Solenoid magnetic field

30.09.2019

The solenoid is a wire wound around a round cylindrical frame. Line B of the solenoid field looks approximately as shown in Fig. 50.1. Inside the solenoid, the direction of these lines forms a right-handed system with the direction of the current in the turns.

A real solenoid has a current component along the axis. In addition, the linear current density (equal to the ratio of the current to the solenoid length element) changes periodically as it moves along the solenoid. The average value of this density is

where is the number of turns of the solenoid per unit of its length, I is the current strength in the solenoid.

In the study of electromagnetism, an important role is played by an imaginary infinitely long solenoid, which has no axial current component and, in addition, the linear current density is constant along its entire length. The reason for this is that the field of such a solenoid is uniform and limited by the volume of the solenoid (similarly, the electric field of an infinite parallel-plate capacitor is uniform and limited by the volume of the capacitor).

In accordance with the above, let us imagine the solenoid in the form of an infinite thin-walled cylinder, flown around by a current of constant linear density

Let's divide the cylinder into identical circular currents - “turns”.

From Fig. 50.2 it can be seen that each pair of turns, located symmetrically relative to a certain plane perpendicular to the axis of the solenoid, creates a magnetic induction parallel to the axis at any point of this plane. Consequently, the resulting field at any point inside and outside the infinite solenoid can only have a direction parallel to the axis.

From Fig. 50.1 it follows that the directions of the field inside and outside the finite solenoid are opposite. As the length of the solenoid increases, the directions of the fields do not change and, in the limit at, remain opposite. For an infinite solenoid, as for a finite one, the direction of the field inside the solenoid forms a right-handed system with the direction of current flow around the cylinder.

From the parallelism of the vector B to the axis, it follows that the field both inside and outside the infinite solenoid must be uniform. To prove this, let's take an imaginary rectangular contour 1-2-3-4 inside the solenoid (Fig. 50.3; the section runs along the axis of the solenoid). By going around the circuit clockwise, we obtain the value for the circulation of vector B. The circuit does not cover currents, so the circulation must be equal to zero (see (49.7)).

It follows that by placing section of circuit 2-3 at any distance from the axis, we will each time obtain that the magnetic induction at this distance is equal to the induction on the axis of the solenoid. Thus, the homogeneity of the field inside the solenoid is proven.

Now let's look at the 1-2-3-4 circuit. We have depicted the vectors with a dashed line, since, as will become clear later, the field outside the infinite solenoid is zero. For now, we only know that the possible direction of the field outside the solenoid is opposite to the direction of the field inside the solenoid. The circuit does not cover currents; therefore, the circulation of vector B along this contour, equal to a, must be equal to zero.

It follows from this that . The distances from the solenoid axis to sections 1-4 and 2-3 were taken arbitrarily. Consequently, the value of B at any distance from the axis will be the same outside the solenoid. Thus, the homogeneity of the field outside the solenoid is also proven.

Circulation along the circuit shown in Fig. 50.4 is equal (for clockwise traversal). This circuit carries a positive current of magnitude . In accordance with (49.7), the equality must be satisfied

or after abbreviation by a and replacement by (see)

From this equality it follows that the field both inside and outside the infinite solenoid is finite.

Let's take a plane perpendicular to the axis of the solenoid (Fig. 50.5). Due to the closedness of lines B, the magnetic fluxes through the inner part 5 of this plane and through the outer part S must be the same.

Since the fields are uniform and perpendicular to the plane, each of the fluxes is equal to the product of the corresponding value of magnetic induction and the area penetrated by the flux. Thus, we get the relation

The left side of this equality is finite, the factor S on the right side is infinitely large. It follows that

So, we have proven that outside an infinitely long solenoid the magnetic induction is zero. The field inside the solenoid is uniform.

Putting in (50.3), we arrive at the formula for the magnetic induction inside the solenoid:

The product is called the number of ampere-turns per meter. With turns per meter and a current of 1 A, the magnetic induction inside the solenoid is .

Symmetrically located turns make an equal contribution to the magnetic induction on the solenoid axis (see formula (47.4)). Therefore, at the end of a semi-infinite solenoid on its axis, the magnetic induction is equal to half the value (50.4): - the number of turns per unit length). In this case

A circuit passing outside the toroid does not cover any current, therefore, for it, thus, outside the toroid the magnetic induction is zero.

For a toroid whose radius R significantly exceeds the radius of the coil, the ratio for all points inside the toroid differs little from unity and instead of (50.6) a formula is obtained that coincides with formula (50.4) for an infinitely long solenoid. In this case, the field can be considered uniform in each section of the toroid. In different sections the field has a different direction, so we can only speak about the homogeneity of the field within its toroid conditionally, bearing in mind the same modulus B.

A real toroid has a current component along its axis. This component creates, in addition to the field (50.6), a field similar to the field of a circular current.

Solenoid called a conductor twisted in a spiral through which an electric current is passed (Figure 1, A).

If you mentally cut the turns of the solenoid across, designate the direction of the current in them, as indicated above, and determine the direction of the magnetic induction lines according to the “gimlet rule”, then the magnetic field of the entire solenoid will have the form as shown in Figure 1, b.

Figure 1. Solenoid ( A) and its magnetic field ( b)

Figure 2. Computer model of the solenoid

On the axis of an infinitely long solenoid, on each unit of length of which is wound n 0 turns, the magnetic field strength inside the solenoid is determined by the formula:

H = I × n 0 .

At the point where the magnetic lines enter the solenoid, a south pole is formed, and where they exit, a north pole is formed.

To determine the poles of the solenoid, they use the “gimlet rule”, applying it as follows: if you place the gimlet along the axis of the solenoid and rotate it in the direction of the current in the turns of the solenoid coil, then the translational movement of the gimlet will show the direction of the magnetic field (Figure 3).

Video about the solenoid:

Electromagnet

A solenoid with a steel (iron) core inside is called electromagnet(Figure 4 and 5). The magnetic field of an electromagnet is stronger than that of a solenoid because a piece of steel inserted into the solenoid is magnetized and the resulting magnetic field is strengthened. The poles of an electromagnet can be determined, just like those of a solenoid, using the “gimlet rule”.

Figure 5. Electromagnet coil

Electromagnets are widely used in technology. They are used to create a magnetic field in electric generators and motors, in electrical measuring instruments, electrical devices and the like.

In high-power installations, instead of fuses, automatic, oil and air circuit breakers are used to disconnect the damaged section of the circuit. Various relays are used to actuate the trip coils of circuit breakers. Relays are devices or machines that respond to changes in current, voltage, power, frequency and other parameters.

From the large number of relays, different in their purpose, principle of operation and design, we will briefly consider the design of electromagnetic relays. Figure 6 shows the designs of these relays. The operation of the relay is based on the interaction of the magnetic field created by a stationary coil through which current passes, and the steel movable armature of an electromagnet. When operating conditions in the main current circuit change, the relay coil is excited, the magnetic flux of the core pulls (turns or retracts) the armature, which closes the contacts of the circuit, the tripping coil of the drive of oil and air switches or auxiliary relays.

Figure 6. Electromagnetic relay

Relays have also found application in automation and telemechanics.

The magnetic flux of a solenoid (electromagnet) increases with the number of turns and current in it. The magnetizing force depends on the product of the current and the number of turns (number of ampere-turns).

If, for example, we take a solenoid whose winding carries a current of 5 A and the number of turns of which is 150, then the number of ampere-turns will be 5 × 150 = 750. The same magnetic flux will be obtained if we take 1500 turns and pass a current of 0.5 through them Ah, since 0.5 × 1500 = 750 ampere-turns.

The magnetic flux of the solenoid can be increased in the following ways: 1) insert a steel core into the solenoid, turning it into an electromagnet; 2) increase the cross-section of the steel core of the electromagnet (since, given the current, magnetic field strength, and therefore magnetic induction, an increase in the cross-section leads to an increase in the magnetic flux); 3) reduce the air gap of the electromagnet core (since when the path of magnetic lines through the air is reduced, the magnetic resistance decreases).

Video about electromagnet:

Solenoid called a cylindrical coil of wire, the turns of which are wound closely in one direction, and the length of the coil is significantly greater than the radius of the turn.

The magnetic field of a solenoid can be represented as the result of the addition of fields created by several circular currents having a common axis. Figure 3 shows that inside the solenoid, the magnetic induction lines of each individual turn have the same direction, while between adjacent turns they have the opposite direction.

Therefore, with a sufficiently dense winding of the solenoid, the oppositely directed sections of the magnetic induction lines of adjacent turns are mutually destroyed, and the equally directed sections will merge into a common magnetic induction line passing inside the solenoid and enveloping it from the outside. Studying this field using sawdust showed that inside the solenoid the field is uniform, the magnetic lines are straight lines parallel to the axis of the solenoid, which diverge at its ends and close outside the solenoid (Fig. 4).

It is easy to notice the similarity between the magnetic field of the solenoid (outside it) and the magnetic field of a permanent bar magnet (Fig. 5). The end of the solenoid from which the magnetic lines come out is similar to the north pole of a magnet N, the other end of the solenoid, into which the magnetic lines enter, is similar to the south pole of the magnet S.

The poles of a current-carrying solenoid can be easily determined experimentally using a magnetic needle. Knowing the direction of the current in the coil, these poles can be determined using the rule of the right screw: we rotate the head of the right screw according to the current in the coil, then the translational movement of the tip of the screw will indicate the direction of the magnetic field of the solenoid, and therefore its north pole. The magnetic induction module inside a single-layer solenoid is calculated by the formula

B = μμ 0 NI l = μμ 0 nl,

Where Ν - number of turns in the solenoid, I— solenoid length, n- the number of turns per unit length of the solenoid.

Magnetization of a magnet. Magnetization vector.

If current flows through a conductor, then an MF is created around the conductor. We have so far looked at wires through which currents flowed in a vacuum. If the wires carrying current are in some medium, then m.p. changes. This is explained by the fact that under the influence of m.p. any substance is capable of acquiring a magnetic moment, or being magnetized (the substance becomes magnetic). Substances that are magnetized in the external mp. against the direction of the field are called diamagnetic materials. Substances that are weakly magnetized in the external magnetic field. in the direction of the field are called paramagnetic materials The magnetized substance creates a magnetic field. - , this is m.p. superimposed on the m.p., caused by currents - . Then the resulting field is:

(54.1)

The true (microscopic) field in a magnet varies greatly within intermolecular distances. - averaged macroscopic field.

For explanation magnetization bodies Ampere suggested that circular microscopic currents circulate in the molecules of a substance, caused by the movement of electrons in atoms and molecules. Each such current has a magnetic moment and creates a magnetic field in the surrounding space.

If there is no external field, then the molecular currents are randomly oriented, and the resulting field due to them is equal to 0.

Magnetization is a vector quantity equal to the magnetic moment of a unit volume of a magnet:

(54.3)

where is a physically infinitesimal volume taken in the vicinity of the point under consideration; - magnetic moment of an individual molecule.

The summation is carried out over all molecules contained in the volume (remember where, - polarization dielectric, - dipole element ).

Magnetization can be represented as follows:

Magnetizing currents I". The magnetization of a substance is associated with the preferential orientation of the magnetic moments of individual molecules in one direction. The elementary circular currents associated with each molecule are called molecular. Molecular currents turn out to be oriented, i.e. magnetizing currents arise - .

Currents flowing through wires due to the movement of current carriers in the substance are called conduction currents -.

For an electron moving in a circular orbit clockwise; the current is directed counterclockwise and, according to the rule of the right screw, is directed vertically upward.

Circulation of the magnetization vector along an arbitrary closed contour is equal to the algebraic sum of the magnetizing currents covered by the contour G.

Differential form of writing the vector circulation theorem.

Magnetic field strength (standard designation N) is a vector physical quantity equal to the difference in the magnetic induction vector B and magnetization vector M.

In SI: Where - magnetic constant.

In the simplest case of an isotropic (in terms of magnetic properties) medium and in the approximation of sufficiently low frequencies of field changes B And H simply proportional to each other, differing simply by a numerical factor (depending on the environment) B = μ H in system GHS or B = μ 0 μ H in system SI(cm. Magnetic permeability, also see Magnetic susceptibility).

In system GHS magnetic field strength is measured in Oerstedach(E), in the SI system - in amperes per meter(A/m). In technology, the oersted is gradually being replaced by the SI unit - ampere per meter.

1 E = 1000/(4π) A/m ≈ 79.5775 A/m.

1 A/m = 4π/1000 Oe ≈ 0.01256637 Oe.

Physical meaning

In a vacuum (or in the absence of a medium capable of magnetic polarization, as well as in cases where the latter is negligible), the magnetic field strength coincides with the magnetic induction vector up to a coefficient equal to 1 in the CGS and μ 0 in the SI.

IN magnets(magnetic environments) the magnetic field strength has the physical meaning of an “external” field, that is, it coincides (perhaps, depending on the adopted units of measurement, to within a constant coefficient, such as in the SI system, which does not change the general meaning) with such a vector magnetic induction, which “would exist if there was no magnet.”

For example, if the field is created by a current-carrying coil into which an iron core is inserted, then the magnetic field strength H inside the core coincides (in GHS exactly, and in SI - up to a constant dimensional coefficient) with the vector B 0, which would be created by this coil in the absence of a core and which, in principle, can be calculated based on the geometry of the coil and the current in it, without any additional information about the material of the core and its magnetic properties.

It should be borne in mind that a more fundamental characteristic of the magnetic field is the magnetic induction vector B . It is he who determines the strength of the magnetic field on moving charged particles and currents, and can also be directly measured, while the magnetic field strength H can be considered rather as an auxiliary quantity (although it is easier to calculate it, at least in the static case, which is where its value lies: after all H create so-called free currents, which are relatively easy to directly measure, while those that are difficult to measure associated currents- that is, molecular currents, etc. - do not need to be taken into account).

True, the commonly used expression for the magnetic field energy (in a medium) B And H enter almost equally, but we must keep in mind that this energy also includes the energy expended on the polarization of the medium, and not just the energy of the field itself. The energy of the magnetic field as such is expressed only through the fundamental B . Nevertheless, it is clear that the value H phenomenologically and here it is very convenient.

Types of magnetic materials Diamagnetic materials have a magnetic permeability of slightly less than 1. They differ in that they are pushed out of the region of the magnetic field.

Paramagnets have a magnetic permeability of slightly more than 1. The overwhelming majority of materials are dia- and paramagnetic.

Ferromagnets have an exceptionally high magnetic permeability, reaching up to a million.

As the field strengthens, the phenomenon of hysteresis appears, when with an increase in intensity and with a subsequent decrease in intensity, the values of B(H) do not coincide with each other. There are several definitions of magnetic permeability in the literature.

Initial magnetic permeability m n- the value of magnetic permeability at low field strength.

Maximum magnetic permeability m max- the maximum value of magnetic permeability, which is usually achieved in average magnetic fields.

Of the other basic terms characterizing magnetic materials, we note the following.

Saturation magnetization- maximum magnetization, which is achieved in strong fields, when all magnetic moments of the domains are oriented along the magnetic field.

Hysteresis loop- dependence of induction on the magnetic field strength when the field changes in a cycle: rise to a certain value - decrease, transition through zero, after reaching the same value with the opposite sign - increase, etc.

Maximum hysteresis loop- reaching maximum saturation magnetization.

Residual induction B rest- magnetic field induction on the reverse stroke of the hysteresis loop at zero magnetic field strength.

Coercive force N s- field strength on the return stroke of the hysteresis loop at which zero induction is achieved.

Magnetic moments of atoms

Magnetic moment Elementary particles have an internal quantum mechanical property known as spin. It is similar to the angular momentum of an object rotating around its own center of mass, although strictly speaking, these particles are point particles and one cannot talk about their rotation. Spin is measured in units of the reduced Planck constant (), then electrons, protons and neutrons have a spin equal to ½. In an atom, electrons orbit the nucleus and have orbital angular momentum in addition to spin, while the nucleus itself has angular momentum due to nuclear spin. The magnetic field created by the magnetic moment of an atom is determined by these different forms of angular momentum, just as in classical physics spinning charged objects create a magnetic field.

However, the most significant contribution comes from spin. Due to the property of the electron, like all fermions, to obey the Pauli exclusion rule, according to which two electrons cannot be in the same quantum state, the bound electrons pair with each other, and one of the electrons is in a spin-up state and the other is spin-up. with the opposite projection of spin - a state with spin down. In this way, the magnetic moments of the electrons are reduced, reducing the total magnetic dipole moment of the system to zero in some atoms with an even number of electrons. In ferromagnetic elements such as iron, an odd number of electrons results in an unpaired electron and a non-zero total magnetic moment. The orbitals of neighboring atoms overlap, and the lowest energy state is achieved when all the spins of the unpaired electrons adopt the same orientation, a process known as exchange interaction. When the magnetic moments of ferromagnetic atoms align, the material can produce a measurable macroscopic magnetic field.

Paramagnetic materials are composed of atoms whose magnetic moments are misoriented in the absence of a magnetic field, but the magnetic moments of individual atoms are aligned when a magnetic field is applied. The nucleus of an atom can also have non-zero total spin. Typically, in thermodynamic equilibrium, nuclear spins are randomly oriented. However, for some elements (such as xenon-129) it is possible to polarize a significant portion of the nuclear spins to create a spin-co-directional state, a state called hyperpolarization. This condition has important applied significance in magnetic resonance imaging.

A magnetic field has energy. Just as there is a reserve of electrical energy in a charged capacitor, there is a reserve of magnetic energy in the coil through which current flows.

If you connect an electric lamp parallel to a coil with high inductance in a direct current electrical circuit, then when the key is opened, a short-term flash of the lamp is observed. 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 W m of the magnetic field of a coil with inductance L created by current I is equal to

W m = LI 2 / 2

A solenoid is a cylindrical coil made of wire, the turns of which are wound in one direction (Fig. 223). The magnetic field of a solenoid is the result of the addition of fields created by several circular currents located nearby and having a common axis.

In Fig. 223 shows four turns of a solenoid with current. For clarity, the half-turns located behind the plane of the sheet are shown as dashed lines. This figure shows that inside the solenoid, the force lines of each individual turn have the same direction, while between adjacent turns they have opposite directions. Therefore, with a sufficiently dense winding of the solenoid, the oppositely directed sections of the force lines of adjacent turns are mutually

will be destroyed, and the equally directed sections will merge into a common closed line of force, passing inside the entire solenoid and enveloping it from the outside.

A detailed study of the magnetic field of a long solenoid, carried out using iron filings, shows that this field has the form shown in Fig. 224. Inside the solenoid the field turns out to be practically uniform, outside the solenoid it is inhomogeneous and relatively weak (the density of the field lines here is very small).

The external field of the solenoid is similar to the field of a bar magnet (see Fig. 212). Like a magnet, a solenoid has a north C pole, a south pole and a neutral zone.

The magnetic field strength inside a long solenoid is calculated by the formula

where I is the length of the solenoid, the number of its turns, and the current strength in it. The product is usually called the number of ampere-turns

Formula (18) is a special case of expressing the field strength inside a solenoid of finite length, which in turn is derived as follows.

In Fig. 225 shows a longitudinal section of the solenoid with a vertical plane passing through its axis. The length of the solenoid I, the radius of its turns, the number of turns, the current strength passing through the solenoid,

Considering the solenoid as a set of turns closely adjacent to each other (circular currents having a common axis, we determine the magnetic field strength at point A on the axis of the solenoid as the sum of the strengths from all its turns. To do this, we select a small section of the length of the solenoid.

It contains turns. According to formula (17), the field strength of one turn Therefore, the field strength from the section will be equal to

From Fig. 225 it is clear that Then Substituting these expressions into

formula (19) and making reductions, we get

Integrating the last expression in the range from to we find the total field strength at point A:

Let us calculate, using the circulation theorem, the magnetic field induction inside solenoid. Consider a solenoid with length l having N turns through which current flows (Fig. 175). We consider the length of the solenoid to be many times greater than the diameter of its turns, i.e. the solenoid in question is infinitely long. Experimental study of the magnetic field of the solenoid (see Fig. 162, b) shows that inside the solenoid the field is uniform, outside the solenoid it is inhomogeneous and very weak.

In Fig. 175 shows the lines of magnetic induction inside and outside the solenoid. The longer the solenoid, the less magnetic induction outside it. Therefore, we can approximately assume that the field of an infinitely long solenoid is concentrated entirely inside it, and the field outside the solenoid can be neglected.

To find magnetic induction IN select a closed rectangular contour ABCDA as shown in fig. 175. Vector circulation IN in a closed loop ABCDA covering everything N turns, according to (118.1), is equal to

Integral over ABCDA can be represented in the form of four integrals: according AB, BC, CD And D.A. At the sites AB And CD the circuit is perpendicular to the lines of magnetic induction and B l = 0. In the area outside the solenoid B=0. Location on D.A. vector circulation IN equal to Bl(the circuit coincides with the magnetic induction line); hence,

(119.1)

From (119.1) we arrive at the expression for the magnetic induction of the field inside the solenoid (in vacuum):

We found that the field inside the solenoid homogeneously(edge effects in areas adjacent to the ends of the solenoid are neglected in calculations). However, we note that the derivation of this formula is not entirely correct (the magnetic induction lines are closed, and the integral over the external portion of the magnetic field is not strictly equal to zero). The field inside the solenoid can be correctly calculated by applying the Biot-Savart-Laplace law; the result is the same formula (119.2).

The magnetic field is also important for practice. toroid- a ring coil, the turns of which are wound on a torus-shaped core (Fig. 176). The magnetic field, as experience shows, is concentrated inside the toroid; there is no field outside it.

The lines of magnetic induction in this case, as follows from symmetry considerations, are circles whose centers are located along the axis of the toroid. As a contour, we choose one such circle of radius r. Then, according to the circulation theorem (118.1), B× 2p r =m 0 NI whence it follows that magnetic induction inside the toroid (in vacuum)

Where N- number of toroid turns.

If the circuit passes outside the toroid, then it does not cover currents and B× 2p r = 0. This means that there is no field outside the toroid (as experience also shows).