Magnet flow. Flux of magnetic field induction. Lenz's rule for magnetic flux

DEFINITION

Flux of magnetic induction vector(or magnetic flux) (dФ) in the general case, through an elementary area is called a scalar physical quantity, which is equal to:

where is the angle between the direction of the magnetic induction vector () and the direction of the normal vector () to the site dS ().

Based on formula (1), the magnetic flux through an arbitrary surface S is calculated (in the general case) as:

magnetic flux uniform magnetic field through a flat surface can be found as:

For a uniform field, a flat surface located perpendicular to the magnetic induction vector, the magnetic flux is equal to:

The flux of the magnetic induction vector can be negative and positive. This is due to the choice of a positive direction. Very often, the flux of the magnetic induction vector is associated with a circuit through which current flows. In this case, the positive direction of the normal to the contour is related to the direction of current flow by the rule of the right gimlet. Then, the magnetic flux, which is created by a current-carrying circuit, through the surface bounded by this circuit, is always greater than zero.

The unit of measure for the flux of magnetic induction in the international system of units (SI) is the weber (Wb). Formula (4) can be used to determine the unit of magnetic flux. One Weber is called a magnetic flux that passes through a flat surface, an area of ​​which 1 square meter, placed perpendicular to the lines of force of a uniform magnetic field:

Gauss theorem for magnetic field

The Gauss theorem for a magnetic field flux reflects the fact that there are no magnetic charges, which is why the lines of magnetic induction are always closed or go to infinity, they have no beginning and end.

The Gauss theorem for the magnetic flux is formulated as follows: The magnetic flux through any closed surface (S) is equal to zero. In mathematical form, this theorem is written as follows:

It turns out that the Gauss theorems for the fluxes of the magnetic induction vector () and the strength of the electrostatic field (), through a closed surface, differ fundamentally.

Examples of problem solving

EXAMPLE 1

Exercise	Calculate the flux of the magnetic induction vector through the solenoid, which has N turns, the length of the core l, the cross-sectional area S, the magnetic permeability of the core. The current flowing through the solenoid is I.
Solution	Inside the solenoid, the magnetic field can be considered uniform. The magnetic induction is easy to find using the magnetic field circulation theorem and choosing a rectangular circuit as a closed circuit (the circulation of the vector along which we will consider (L)) a rectangular circuit (it will cover all N turns). Then we write (we take into account that outside the solenoid the magnetic field is zero, in addition, where the contour L is perpendicular to the lines of magnetic induction B = 0): In this case, the magnetic flux through one turn of the solenoid is (): The total flux of magnetic induction that goes through all the turns:
Answer

EXAMPLE 2

Exercise	What will be the flux of magnetic induction through a square frame, which is in vacuum in the same plane with an infinitely long straight conductor with current (Fig. 1). The two sides of the frame are parallel to the wire. The length of the side of the frame is b, the distance from one of the sides of the frame is c.
Solution	The expression by which it is possible to determine the induction of the magnetic field will be considered known (see Example 1 of the section "Magnetic induction unit of measure"):

What is magnetic flux?

The picture shows a uniform magnetic field. Homogeneous means the same at all points in a given volume. A surface with area S is placed in the field. Field lines intersect the surface.

Magnetic flux definition

Definition of magnetic flux:

The magnetic flux Ф through the surface S is the number of lines of the magnetic induction vector B passing through the surface S.

Magnetic flux formula

Magnetic flux formula:

here α is the angle between the direction of the magnetic induction vector B and the normal to the surface S.

It can be seen from the magnetic flux formula that the maximum magnetic flux will be at cos α = 1, and this will happen when the vector B is parallel to the normal to the surface S. The minimum magnetic flux will be at cos α = 0, this will be when the vector B is perpendicular to the normal to the surface S, because in this case the lines of the vector B will slide over the surface S without crossing it.

And according to the definition of the magnetic flux, only those lines of the magnetic induction vector that intersect a given surface are taken into account.

Magnetic flux is a scalar quantity.

The magnetic flux is measured

The magnetic flux is measured in webers (volt-seconds): 1 wb \u003d 1 v * s.

In addition, Maxwell is used to measure the magnetic flux: 1 wb \u003d 10 8 μs. Accordingly, 1 μs = 10 -8 wb.

The picture shows a uniform magnetic field. Homogeneous means the same at all points in a given volume. A surface with area S is placed in the field. Field lines intersect the surface.

Determination of magnetic flux:

The magnetic flux Ф through the surface S is the number of lines of the magnetic induction vector B passing through the surface S.

Magnetic flux formula:

here α is the angle between the direction of the magnetic induction vector B and the normal to the surface S.

It can be seen from the magnetic flux formula that the maximum magnetic flux will be at cos α = 1, and this will happen when the vector B is parallel to the normal to the surface S. The minimum magnetic flux will be at cos α = 0, this will be when the vector B is perpendicular to the normal to the surface S, because in this case the lines of the vector B will slide over the surface S without crossing it.

And according to the definition of the magnetic flux, only those lines of the magnetic induction vector that intersect a given surface are taken into account.

The magnetic flux is measured in webers (volt-seconds): 1 wb \u003d 1 v * s. In addition, Maxwell is used to measure the magnetic flux: 1 wb \u003d 10 8 μs. Accordingly, 1 μs = 10 -8 wb.

Magnetic flux is a scalar quantity.

ENERGY OF THE MAGNETIC FIELD OF THE CURRENT

Around a conductor with current there is a magnetic field that has energy. Where does it come from? The current source included in the electric circuit has an energy reserve. At the moment of closing the electric circuit, the current source expends part of its energy to overcome the action of the emerging EMF of self-induction. This part of the energy, called the self-energy of the current, goes to the formation of a magnetic field. The energy of the magnetic field is equal to the self-energy of the current. The self-energy of the current is numerically equal to the work that the current source must do to overcome the self-induction EMF in order to create a current in the circuit.

The energy of the magnetic field created by the current is directly proportional to the square of the current strength. Where does the energy of the magnetic field disappear after the current stops? - stands out (when a circuit with a sufficiently large current is opened, a spark or arc may occur)

4.1. The law of electromagnetic induction. Self-induction. Inductance

Basic Formulas

The law of electromagnetic induction (Faraday's law):

, (39)

where is the induction emf; is the total magnetic flux (flux linkage).

The magnetic flux created by the current in the circuit,

where is the inductance of the circuit; is the current strength.

Faraday's law as applied to self-induction

The emf of induction that occurs when the frame rotates with current in a magnetic field,

where is the magnetic field induction; is the frame area; is the angular velocity of rotation.

solenoid inductance

, (43)

where is the magnetic constant; is the magnetic permeability of the substance; is the number of turns of the solenoid; is the sectional area of ​​the turn; is the length of the solenoid.

Open circuit current

where is the current strength established in the circuit; is the inductance of the circuit; is the resistance of the circuit; is the opening time.

The current strength when the circuit is closed

. (45)

Relaxation time

Examples of problem solving

Example 1

The magnetic field changes according to the law , where = 15 mT,. A circular conducting coil with a radius = 20 cm is placed in a magnetic field at an angle to the direction of the field (at the initial moment of time). Find the emf of induction that occurs in the coil at time = 5 s.

Solution

According to the law of electromagnetic induction, the emf of induction arising in the coil, where is the magnetic flux coupled in the coil.

where is the area of ​​the coil,; is the angle between the direction of the magnetic induction vector and the normal to the contour:.

Substitute the numerical values: = 15 mT,, = 20 cm = = 0.2 m,.

Calculations give .

Example 2 In a uniform magnetic field with an induction = 0.2 T, a rectangular frame is located, the movable side of which is 0.2 m long and moves at a speed of = 25 m/s perpendicular to the field induction lines (Fig. 42). Determine the emf of induction that occurs in the circuit. Solution When the conductor AB moves in a magnetic field, the area of ​​\u200b\u200bthe frame increases, therefore, the magnetic flux through the frame increases and an emf of induction occurs.

According to Faraday's law, where, then, but, therefore.

The “–” sign indicates that the induction emf and the induction current are directed counterclockwise.

SELF-INDUCTION

Each conductor through which electric current flows is in its own magnetic field.

When the current strength changes in the conductor, the m.field changes, i.e. the magnetic flux created by this current changes. A change in the magnetic flux leads to the emergence of a vortex electric field and an induction EMF appears in the circuit. This phenomenon is called self-induction. Self-induction is the phenomenon of induction EMF in an electric circuit as a result of a change in current strength. The resulting emf is called the self-induction emf.

Manifestation of the phenomenon of self-induction

Closing the circuit When a circuit is closed, the current increases, which causes an increase in the magnetic flux in the coil, a vortex electric field arises, directed against the current, i.e. an EMF of self-induction occurs in the coil, which prevents the current from rising in the circuit (the vortex field slows down the electrons). As a result L1 lights up later, than L2.

Open circuit When the electric circuit is opened, the current decreases, there is a decrease in the m.flow in the coil, a vortex electric field appears, directed like a current (tending to maintain the same current strength), i.e. A self-inductive emf appears in the coil, which maintains the current in the circuit. As a result, L when turned off flashes brightly. Conclusion in electrical engineering, the phenomenon of self-induction manifests itself when the circuit is closed (the electric current increases gradually) and when the circuit is opened (the electric current does not disappear immediately).

INDUCTANCE

What does the EMF of self-induction depend on? Electric current creates its own magnetic field. The magnetic flux through the circuit is proportional to the magnetic field induction (Ф ~ B), the induction is proportional to the current strength in the conductor (B ~ I), therefore the magnetic flux is proportional to the current strength (Ф ~ I). The self-induction emf depends on the rate of change in the current strength in the electric circuit, on the properties of the conductor (size and shape) and on the relative magnetic permeability of the medium in which the conductor is located. A physical quantity showing the dependence of the self-induction EMF on the size and shape of the conductor and on the environment in which the conductor is located is called the self-induction coefficient or inductance. Inductance - physical. a value numerically equal to the EMF of self-induction that occurs in the circuit when the current strength changes by 1 ampere in 1 second. Also, the inductance can be calculated by the formula:

where F is the magnetic flux through the circuit, I is the current strength in the circuit.

SI units for inductance:

The inductance of the coil depends on: the number of turns, the size and shape of the coil, and the relative magnetic permeability of the medium (a core is possible).

SELF-INDUCTION EMF

EMF of self-induction prevents the increase in current strength when the circuit is turned on and the decrease in current strength when the circuit is opened.

To characterize the magnetization of a substance in a magnetic field, we use magnetic moment (P m ). It is numerically equal to the mechanical moment experienced by a substance in a magnetic field with an induction of 1 T.

The magnetic moment of a unit volume of a substance characterizes it magnetization - I , is determined by the formula:

I=R m /V , (2.4)

where V is the volume of the substance.

Magnetization in the SI system is measured, like tension, in A/m, the quantity is vector.

The magnetic properties of substances are characterized bulk magnetic susceptibility - c O , the quantity is dimensionless.

If a body is placed in a magnetic field with induction V 0 , then magnetization occurs. As a result, the body creates its own magnetic field with induction V " , which interacts with the magnetizing field.

In this case, the induction vector in the environment (V) will be composed of vectors:

B = B 0 + V " (vector sign omitted), (2.5)

where V " - induction of the own magnetic field of the magnetized substance.

The induction of its own field is determined by the magnetic properties of the substance, which are characterized by volumetric magnetic susceptibility - c O , the expression is true: V " = c O V 0 (2.6)

Divide by m 0 expression (2.6):

V " /m O = c O V 0 /m 0

We get: H " = c O H 0 , (2.7)

but H " determines the magnetization of a substance I , i.e. H " = I , then from (2.7):

I=c O H 0 . (2.8)

Thus, if the substance is in an external magnetic field with a strength H 0 , then inside it the induction is defined by the expression:

B=B 0 + V " = m 0 H 0 +m 0 H " = m 0 (H 0 +I)(2.9)

The last expression is strictly valid when the core (substance) is completely in an external uniform magnetic field (a closed torus, an infinitely long solenoid, etc.).

Ampère's law is used to establish the unit of current strength - ampere.

Ampere - the strength of the current of constant magnitude, which, passing through two parallel rectilinear conductors of infinite length and negligible cross section, located at a distance of one meter, one from the other in a vacuum, causes a force between these conductors.

,

(2.4.1)

Here ; ; ;

From here we determine the dimension and magnitude in SI.

, hence

, or .

From the Biot-Savart-Laplace law, for a rectilinear conductor with current , too one can find the dimension of the magnetic field induction:

Tesla is the SI unit of measurement for induction. .

Gauss- a unit of measure in the Gaussian system of units (CGS).

1 T is equal to the magnetic induction of a uniform magnetic field, in which on a flat circuit with a current having a magnetic moment,torque is applied.

Tesla Nikola(1856–1943) Serbian scientist in the field of electrical and radio engineering. Had great amount inventions. Invented an electric meter, frequency meter, etc. Developed a number of designs of multi-phase generators, electric motors and transformers. He designed a number of radio-controlled self-propelled mechanisms. Studied the physiological effect of high frequency currents. In 1899 he built a 200 kW radio station in Colorado and a 57.6 m high radio antenna in Long Island (Wordenclyffe tower). Together with Einstein and Oppenheimer in 1943, he participated in a secret project to achieve the invisibility of American ships (Philadelphia experiment). Contemporaries spoke of Tesla as a mystic, clairvoyant, prophet, able to look into the intelligent cosmos and the world of the dead. He believed that with the help of an electromagnetic field, one could move in space and control time.

Other definition: 1 T is equal to the magnetic induction at which the magnetic flux through the area 1 m 2, perpendicular to the direction of the field,equals 1 Wb .

The unit of measurement of the magnetic flux, Wb, got its name in honor of the German physicist Wilhelm Weber (1804–1891), a professor at the universities in Halle, Göttingen, and Leipzig.

As we said before magnetic flux Ф through the surface S is one of the characteristics of the magnetic field(Fig. 2.5):

Unit of measurement of magnetic flux in SI:

. , and since , then .

Here Maxwell(Mks) is a CGS unit of magnetic flux named after the famous English scientist James Maxwell (1831–1879), the creator of the theory of the electromagnetic field.

Magnetic field strength H measured in .

, .

Let us summarize in one table the main characteristics of the magnetic field.

Table 2.1

Name

Magnetic materials are those that are subject to the influence of special force fields, in turn, non-magnetic materials are not subject to or weakly subject to the forces of a magnetic field, which is usually represented by lines of force (magnetic flux) that have certain properties. In addition to always forming closed loops, they behave as if they are elastic, that is, during the distortion, they try to return to their previous distance and to their natural shape.

invisible force

Magnets tend to attract certain metals, especially iron and steel, as well as nickel, nickel, chromium and cobalt alloys. Materials that create attractive forces are magnets. There are various types. Materials that can be easily magnetized are called ferromagnetic. They can be hard or soft. Soft ferromagnetic materials such as iron lose their properties quickly. Magnets made from these materials are called temporary. Rigid materials such as steel hold their properties much longer and are used as permanent materials.

Magnetic Flux: Definition and Characterization

Around the magnet there is a certain force field, and this creates the possibility of energy. The magnetic flux is equal to the product of the average force fields of the perpendicular surface into which it penetrates. It is depicted using the symbol "Φ", it is measured in units called Webers (WB). The amount of flow passing through given area, will vary from one point to another around the subject. Thus, magnetic flux is a so-called measure of the strength of a magnetic field or electric current, based on the total number of charged lines of force passing through a certain area.

Revealing the mystery of magnetic fluxes

All magnets, regardless of their shape, have two areas, called poles, capable of producing a certain chain of organized and balanced system of invisible lines of force. These lines from the stream form a special field, the form of which is more intense in some parts than in others. The areas with the greatest attraction are called poles. Vector field lines cannot be detected with the naked eye. Visually, they always appear as lines of force with unambiguous poles at each end of the material, where the lines are denser and more concentrated. Magnetic flux is lines that create vibrations of attraction or repulsion, showing their direction and intensity.

Magnetic flux lines

Magnetic lines of force are defined as curves that move along a certain path in a magnetic field. The tangent to these curves at any point shows the direction of the magnetic field in it. Specifications:

Each flow line forms a closed loop.

These induction lines never intersect, but tend to shrink or stretch, changing their dimensions in one direction or another.

As a rule, lines of force have a beginning and an end on the surface.

There is also a certain direction from north to south.

Field lines that are close to each other, forming a strong magnetic field.

• When adjacent poles are the same (north-north or south-south), they repel each other. When neighboring poles are not aligned (north-south or south-north), they are attracted to each other. This effect is reminiscent of the famous expression that opposites attract.

Magnetic molecules and Weber's theory

Weber's theory relies on the fact that all atoms are magnetic due to the bonds between the electrons in the atoms. Groups of atoms join together in such a way that the fields surrounding them rotate in the same direction. These kinds of materials are made up of groups of tiny magnets (when viewed at the molecular level) around atoms, which means that the ferromagnetic material is made up of molecules that have attractive forces. They are known as dipoles and are grouped into domains. When the material is magnetized, all the domains become one. A material loses its ability to attract and repel when its domains are separated. The dipoles together form a magnet, but individually, each of them tries to repel the unipolar one, thus attracting opposite poles.

Fields and poles

The strength and direction of the magnetic field is determined by the magnetic flux lines. The area of ​​attraction is stronger where the lines are close to each other. The lines are closest to the pole of the rod base, where the attraction is strongest. The planet Earth itself is in this powerful force field. It acts as if a giant striped magnetized plate is running through the middle of the planet. The north pole of the compass needle is directed towards a point called the North magnetic pole, the south pole it points to the magnetic south. However, these directions differ from the geographic North and South Poles.

The nature of magnetism

Magnetism plays an important role in electrical and electronic engineering, because without its components such as relays, solenoids, inductors, chokes, coils, loudspeakers, electric motors, generators, transformers, electricity meters, etc. will not work. Magnets can be found in natural state in the form of magnetic ores. There are two main types, these are magnetite (also called iron oxide) and magnetic ironstone. The molecular structure of this material in a non-magnetic state is presented as a loose magnetic circuit or individual tiny particles that are freely arranged in a random order. When a material is magnetized, this random arrangement of molecules changes, and tiny random molecular particles line up in such a way that they produce a whole series of arrangements. This idea of ​​molecular alignment of ferromagnetic materials is called Weber's theory.

Measurement and practical application

The most common generators use magnetic flux to generate electricity. Its strength is widely used in electrical generators. The device that measures this interesting phenomenon is called a fluxmeter, it consists of a coil and electronic equipment that evaluates the change in voltage in the coil. In physics, a flow is an indicator of the number of lines of force passing through a certain area. Magnetic flux is a measure of the number of magnetic lines of force.

Sometimes even a non-magnetic material can also have diamagnetic and paramagnetic properties. An interesting fact is that the forces of attraction can be destroyed by heating or being struck with a hammer of the same material, but they cannot be destroyed or isolated by simply breaking a large specimen in two. Each broken piece will have its own north and south pole, no matter how small the pieces are.