magnetic tension. Basic Formulas

What is magnetic tension?

Magnetic voltage definition

Determination of magnetic voltage:

The magnetic voltage on a straight section of the contour is the product of the length of the section and the projection of the magnetic intensity vector on this straight section.

All this applies to a uniform magnetic field. If the field is not uniform or the contour section is not straight, then a small part of the contour is selected, which can be considered rectilinear, and the magnetic field at the location of this section is homogeneous.

Magnetic voltage formula

The picture above shows a uniform magnetic field with a strength vector H and a curvilinear contour L. The contour is curvilinear, therefore it is impossible to determine the magnetic voltage on the entire contour at once. Let us single out the segment ΔL on the contour (shown in bold line), which can be considered straight, and we will find the magnetic voltage only in this section. Tension vector projection magnetic field H to the direction of the segment ΔL is equal to:

H L = H * cos α

where α is the angle between the vector H and the segment ΔL.

Magnetic voltage on segment ΔL (magnetic voltage formula):

U m = (H * cos α) * ΔL = H L * ΔL

Highlighting straight sections on the remaining parts of the circuit L, we find the magnetic voltages on them. Then the total magnetic voltage on the entire circuit L will be equal to the sum of the magnetic voltages of the sections:

U L = Σ H L * ΔL

The magnetic voltage is measured in amperes: A.

The magnetic voltage along the contour L depends on the shape of this contour.

Problem about magnetic stress

Now let's decide a simple task: how will the magnetic stresses be related on the segments ΔL, ΔL 1 , ΔL 2 (see figure), i.e. where are they more and where are they less? The lengths of all sections are the same, the magnetic field is uniform everywhere.

Solution. Under these conditions, the magnetic stresses on the indicated segments will differ only in the magnitudes of the projections of the magnetic field strength vector onto the directions of these segments. The segment ΔL 1 is located at a smaller angle to the direction of the vector Η compared to the segments ΔL and ΔL 2, which means cos α is closer to unity and the magnetic voltage there will be greater. The segment ΔL 2 is located at right angles to the direction of the intensity vector, which means that the projection of the intensity vector Η onto the direction of the segment ΔL 2 will be equal to zero.

And now attention, the correct answer: we will get the largest magnetic voltage on the segment ΔL 1, and the smallest - on the segment ΔL 2.

The magnetic field of a permanent magnet is caused by the movement of electrons in their orbits in an atom.

The magnetic field is characterized by intensity. The intensity H of the magnetic field is similar to the mechanical force. It is a vector quantity, that is, it has a magnitude and a direction.

The magnetic field, i.e., the space around the magnet, can be represented as filled with magnetic lines, which are usually considered to be coming out of north pole magnet and entering the southern one (Fig. 1). The tangents to the magnetic line show the direction of the magnetic field strength.

The magnetic field strength is greater where magnetic lines thicker (on the poles of a magnet or inside a coil with current).

The magnetic field near the conductor (or inside the coil) is the greater, the greater the current I and the number of turns ω of the coil.

The strength of the magnetic field H at any point in space is the greater, the greater the product I∙ω and the shorter the length of the magnetic line:

H=(I∙ω)/l.

It follows from the equation that the unit of measurement for magnetic field strength is ampere per meter (A/m).

For each magnetic line in a given homogeneous field, the products H1∙l1=H2∙l2=...=H∙l=I∙ω are equal (Fig. 1).

Rice. one.

The product H∙l in magnetic circuits is similar to the voltage in electrical circuits and is called magnetic voltage, and taken along the entire length of the magnetic induction line is called the magnetizing force (n.s.) Fm: Fm=H∙l=I∙ω.

The magnetizing force Fm is measured in amperes, but in technical practice, instead of the name ampere, the name ampere-turn is used, which emphasizes that Fm is proportional to the current and the number of turns.

For a cylindrical coil without a core, the length of which is much greater than its diameter (l≫d), the magnetic field inside the coil can be considered homogeneous, i.e. having the same magnetic field strength H throughout the entire internal space of the coil (Fig. 1). Since the magnetic field outside such a coil is much weaker than inside it, the external magnetic field can be neglected and, in the calculation, we can assume that n. With. coil is equal to the product of the field strength inside the coil and the length of the coil.

The polarity of the magnetic field of a wire and a current-carrying coil is determined by the gimlet rule. If a forward movement gimlet coincides with the direction of the current, then the direction of rotation of the gimlet handle will indicate the direction of the magnetic lines.

Examples

1. A current of 3 A flows through a coil having 2000 turns. What is n. With. coils?

Fm=I∙ω=3∙2000=6000 A. The magnetizing force of the coil is 6000 ampere-turns.

2. A coil having 2500 turns must have n. With. 10000 A. What current should flow through it?

I=Fm/ω=(I∙ω)/ω=10000/2500=4 A.

3. A current flows through the coil I \u003d 2 A. How many turns should be in the coil to ensure n. With. 8000 Huh?

ω= Fm/I=(I∙ω)/I=8000/2=4000 turns.

4. Inside the coil 10 cm long, having 100 turns, it is necessary to ensure the magnetic field strength H=4000 A/m. What current should flow through the coil?

Magnetizing force of the coil Fм=H∙l=I∙ω. Hence 4000 A / m ∙ 0.1 m = I ∙ 100; I=400/100=4 A.

5. The diameter of the coil (solenoid) is D=20 mm, and its length is l=10 cm. The coil is wound from a copper wire with a diameter of d=0.4 mm. What is the strength of the magnetic field inside the coil if it is turned on at a voltage of 4.5 V?

The number of turns without taking into account the thickness of the insulation ω=l∶d=100∶0.4=250 turns.

The length of the turn π∙d=3.14∙0.02 m = 0.0628 m.

Coil wire length l1=250∙0.0628 m =15.7 m.

Coil active resistance r=ρ∙l1/S=0.0175∙(4∙15.7)/(3.14∙0.16)=2.2 Ohm.

Current I \u003d U / r \u003d 4.5 / 2.2 \u003d 2.045 A ≈2 A.

The magnetic field strength inside the coil H=(I∙ω)/l=(2∙250)/0.1=5000 A/m.

6. Determine the strength of the magnetic field at a distance of 1, 2, 5 cm from a straight wire through which a current flows I = 100 A.

Let's use the formula H∙l=I∙ω.

For a straight wire ω=1 and l=2∙π∙r,

whence H= I/(2∙π∙r).

H1=100/(2∙3.14∙0.01)=1590 A/m; H2=795 A/m; H3=318 A/m.

Good day. In I talked about the main characteristic of the magnetic field - magnetic induction, however, the calculation formulas given correspond to the magnetic field in vacuum. What in practical activities is quite rare. When they are in any environment, even in the air, the magnetic field that they create undergoes some, and sometimes significant changes. What changes occur with the magnetic field, and what it depends on, I will tell in this article.

How are induction and magnetic field strength related?

A magnet is a substance that, under the influence of a magnetic field, is capable of being magnetized (or, as physicists say, acquire a magnetic moment). Almost all substances are magnets. The magnetization of substances is explained by the fact that substances have their own microscopic magnetic fields, which are created by the rotation of electrons in their orbits. When the external is absent, then the microscopic fields are arranged in an arbitrary way, and under the influence of an external magnetic field they are oriented accordingly.

To characterize the magnetization various substances use the so-called magnetization vector J.

Thus, under the action of an external magnetic field with magnetic induction At 0, the magnet is magnetized and creates its own magnetic field with magnetic induction AT'. As a result, the general induction AT will consist of two terms

Here the problem arises of calculating the magnetic induction of a magnetized substance AT', for the solution of which it is necessary to consider the electron microcurrents of the entire substance, which is practically unrealistic.

An alternative to this solution is the input of auxiliary parameters, namely the magnetic field strength H and magnetic susceptibility χ . Tension links magnetic induction AT and magnetization of matter J following expression

where in - magnetic induction,

μ 0 - magnetic constant, μ 0 \u003d 4π * 10 -7 H / m.

At the same time, the magnetization vector J related to magnetic field strength AT a parameter that characterizes the magnetic properties of a substance and is called magnetic susceptibility χ

where J is the substance magnetization vector,

However, most often to characterize magnetic properties substances use the relative magnetic permeability μ r .

Thus, the relationship between intensity and magnetic induction will have the following form

where μ 0 is the magnetic constant, μ 0 = 4π*10 -7 H/m,

μ r is the relative magnetic permeability of the substance.

Since vacuum magnetization is equal to zero (J = 0), then the magnetic field strength in vacuum will be equal to

From here it is possible to derive expressions for the intensity for the magnetic field created by a direct wire with current:

where I is the current flowing through the conductor,

b is the distance from the center of the wire to the point at which the magnetic field strength is considered.

As can be seen from this expression, the unit of measurement of tension is ampere per meter ( A/m) or oersted ( E)

So the magnetic induction AT and tension H are the main characteristics of the magnetic field, and the magnetic permeability μ r- the magnetic characteristic of the substance.

Magnetization of ferromagnets

Depending on the magnetic properties, that is, the ability to be magnetized under the influence of an external magnetic field, all substances are divided into several classes. Which are characterized by different values ​​of relative magnetic permeability μ r and magnetic susceptibility χ. Most substances are diamagnets(χ = -10 -8 … -10 -7 and μ r< 1) и paramagnets (χ \u003d 10 -7 ... 10 -6 and μ r\u003e 1), are somewhat less common ferromagnets(χ = 10 3 ... 10 5 and μ r >> 1). In addition to these classes of magnets, there are several more classes of magnets: antiferromagnets, ferrimagnets, and others, but their properties appear only under certain conditions.

Of particular interest in radio electronics are ferromagnetic substances. The main difference this class substances is a nonlinear dependence of magnetization, in contrast to para- and diamagnets, which have a linear dependence of magnetization J from tension H magnetic field.

Magnetization dependence J ferromagnet against tension H magnetic field.

This chart shows main magnetization curve ferromagnet. Initially, the magnetization J, in the absence of a magnetic field (H = 0), is equal to zero. As the intensity increases, the magnetization of a ferromagnet proceeds quite intensively, due to the fact that its magnetic susceptibility and permeability are very high. However, when the magnetic field strength reaches the order of H ≈ 100 A/m, the increase in magnetization stops, since the saturation point J NAS is reached. This phenomenon is called magnetic saturation. In this mode, the magnetic permeability of ferromagnets drops sharply and tends to unity with a further increase in the magnetic field strength.

Hysteresis of ferromagnets

Another feature of ferromagnets is the presence, which is a fundamental property of ferromagnets.

To understand the process of magnetization of a ferromagnet, we depict the dependence of the induction AT from tension H magnetic field, where we highlight in red main magnetization curve. This dependence is rather uncertain, since it depends on the previous magnetization of the ferromagnet.

Let's take a sample of a ferromagnetic substance that was not subjected to magnetization (point 0) and place it in a magnetic field, the intensity H which we will begin to increase, that is, the dependence will correspond to the curve 0 – 1 until magnetic saturation is reached (point 1). A further increase in tension does not make sense, because the magnetization J practically does not increase, and the magnetic induction increases in proportion to the intensity H. If we begin to reduce tension, then dependence H(H) will fit the curve 1 – 2 – 3 , while when the magnetic field strength drops to zero (point 2), then the magnetic induction will not drop to zero, but will be equal to a certain value B r, which is called residual induction, and the magnetization will matter J r called residual magnetization.

In order to remove residual magnetization and reduce residual induction B r to zero, it is necessary to create a magnetic field opposite to the field that caused the magnetization, and the intensity of the demagnetizing field should be N s called coercive force. With a further increase in the strength of the magnetic field, which is opposite to the initial field, the saturation of the ferromagnet occurs (point 4).

Thus, when an alternating magnetic field acts on a ferromagnet, the dependence of the induction on the intensity will correspond to the curve 1 – 2 – 3 – 4 – 5 – 6 – 1 , which is called hysteresis loop. There can be many such loops for a ferromagnet (dashed curves), called private cycles. However, if at maximum values magnetic field strength is saturated, it turns out maximum hysteresis loop(solid curve).

Since the magnetic permeability μ r of ferromagnets has a rather complex dependence on the magnetic field strength, therefore, two parameters of the magnetic permeability are normalized:

μ n - the initial magnetic permeability corresponds to the intensity H = 0;

μ max - the maximum magnetic permeability is reached in a magnetic field when magnetic saturation is approaching.

Thus, for ferromagnets, the values ​​of Br, H c and μ n (μ max) are the main characteristics that affect the choice of substance in a particular case.

Theory is good, but practical application it's just words.

In magnets (magnetic media), the magnetic field strength is physical meaning"external" field, that is, it coincides (perhaps, depending on the accepted units of measurement, up to constant coefficient, such as in the SI system, which does not change the general meaning) with such a magnetic induction vector, which “would be if there was no magnet”.

For example, if the field is created by a coil with current into which an iron core is inserted, then the magnetic field strength H inside the core coincides (in CGS 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.

At the same time, 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 its value: after all, H create the so-called free currents, which are relatively easy to directly measure, but difficult to measure coupled currents- that is, molecular currents, etc. - do not need to be taken into account).

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

1. The torque acting on the frame with current from the side of the magnetic field. Magnetic moment of the frame with current. Torque. Determination of magnetic field induction. Units of induction and torque.

By placing the frame in a uniform magnetic field, a couple of forces act on it, which creates a torque.

2. Magnetic field strength and its connection with induction. Tension unit.

The magnetic induction vector is a general characteristic of the points of the magnetic field, regardless of how the magnetic field is created: by a magnetized body or a conductor with current in a given medium.

However, it is possible to introduce some characteristic of the magnetic field that does not depend on the medium, but is determined by the currents and the configuration of the conductors - magnetic field strength vector. These two characteristics (one general and the other particular) are related: where - absolute magnetic permeability of vacuum, μ - relative magnetic permeability of the medium, for vacuum μ = 1.

Magnetic field strength- the ratio of the mechanical force acting on the positive pole of the test magnet to the value of its magnetic mass or the mechanical force acting on the positive pole of the test magnet of unit mass at a given point of the field.

Unit of magnetic field strength- ampere per meter (A / m): 1 A / m - the intensity of such a field, the magnetic induction of which in vacuum is 4π * T.

3. Image of magnetic fields using induction lines of force (strength). Type of lines of magnetic induction of direct and circular currents, solenoid. Rules, but which determine the direction of the lines of magnetic induction.

4. Magnetic fields of conductors with currents. Biot-Savart-Laplace law.

A magnetic field- this is a force field acting on moving electric charges and on bodies with a magnetic moment, regardless of the state of their movement.

Biot-Savart-Laplace law:

In vector form:

In scalar form:

5. Application of the Biot-Savart-Laplace law to determine the field strength generated by:

a) a straight conductor of finite length (derivation of the formula)

b) an infinitely long straight conductor (derivation of the formula)

c) a circular conductor in the center (derivation of the formula)

d) solenoid and toroid

e) circular conductor on the axis (without output)

6. Power of Ampere. Rule for determining the direction of Ampère's force.

A current-carrying conductor in a magnetic field is subjected to a force equal to F = I L B sina

I - current strength in the conductor; B is the modulus of the magnetic field induction vector; L is the length of the conductor in the magnetic field; a is the angle between the magnetic field vector and the direction of the current in the conductor.

Amp power- The force acting on a current-carrying conductor in a magnetic field.

The maximum force of Ampere is: F = I·L·B. It corresponds to a = 90.

The direction of the Ampère force is determined left hand rule: if left hand position so that the perpendicular component of the magnetic induction vector B enters the palm, and four outstretched fingers are directed in the direction of the current, then bent 90 degrees thumb will show the direction of the force acting on a piece of conductor with current, that is, the Ampère force.