What physical quantity is called magnetic flux. Magnetic field induction flux

Among physical quantities, an important place is occupied by magnetic flux. This article explains what it is and how to determine its value.

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Magnetic flux formula

What is magnetic flux

This is the value that determines the level magnetic field passing through the surface. Denoted "FF" and depends on the strength of the field and the angle of passage of the field through this surface.

It is calculated according to the formula:

FF=B⋅S⋅cosα, where:

FF - magnetic flux;
B is the value of magnetic induction;
S is the surface area through which this field passes;
cosα is the cosine of the angle between the perpendicular to the surface and the flow.

The SI unit of measure is "weber" (Wb). 1 weber is created by a 1 T field passing perpendicular to a surface of 1 m².

Thus, the flow is maximum when its direction coincides with the vertical and is equal to "0" if it is parallel to the surface.

Interesting. The formula for the magnetic flux is similar to the formula by which the illumination is calculated.

permanent magnets

One of the sources of the field are permanent magnets. They have been known for centuries. The compass needle was made of magnetized iron, and in Ancient Greece there was a legend about an island that attracted the metal parts of ships to itself.

There are permanent magnets various shapes and are made from different materials:

iron - the cheapest, but have less attractive force;
neodymium - from an alloy of neodymium, iron and boron;
Alnico is an alloy of iron, aluminum, nickel and cobalt.

All magnets are bipolar. This is most noticeable in rod and horseshoe devices.

If the rod is hung in the middle or placed on a floating piece of wood or foam, then it will turn in the north-south direction. The pole pointing north is called the north pole and is painted in laboratory instruments. blue color and denoted by "N". The opposite one, pointing south, is red and marked "S". Like poles attract magnets, while opposite poles repel.

In 1851, Michael Faraday proposed the concept of closed lines of induction. These lines leave the north pole of the magnet, pass through the surrounding space, enter the south and inside the device return to the north. The closest lines and field strengths are near the poles. Here, too, the attraction force is higher.

If you put a piece of glass on the device, and on top thin layer pour iron filings, then they will be located along the lines of the magnetic field. When several devices are located next to each other, the sawdust will show the interaction between them: attraction or repulsion.

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Magnet and iron filings

Earth's magnetic field

Our planet can be represented as a magnet, the axis of which is tilted by 12 degrees. The intersections of this axis with the surface are called magnetic poles. Like any magnet, the Earth's lines of force run from the north pole to the south. Near the poles, they run perpendicular to the surface, so the compass needle is unreliable there, and other methods have to be used.

The particles of the "solar wind" have an electric charge, so when moving around them, a magnetic field appears that interacts with the Earth's field and directs these particles along the lines of force. Thus, this field protects the earth's surface from cosmic radiation. However, near the poles, these lines are perpendicular to the surface, and charged particles enter the atmosphere, causing the aurora borealis.

electromagnets

In 1820, Hans Oersted, while conducting experiments, saw the effect of a conductor through which electricity, on the compass needle. A few days later, André-Marie Ampere discovered the mutual attraction of two wires, through which a current flowed in the same direction.

Interesting. During electric welding, nearby cables move when the current changes.

Ampère later suggested that this was due to the magnetic induction of the current flowing through the wires.

In a coil wound with an insulated wire through which an electric current flows, the fields of the individual conductors reinforce each other. To increase the attractive force, the coil is wound on an open steel core. This core becomes magnetized and attracts iron parts or the other half of the core in relays and contactors.

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electromagnets

Electromagnetic induction

When the magnetic flux changes, an electric current is induced in the wire. This fact does not depend on what causes this change was caused: the displacement permanent magnet, the movement of a wire or a change in the strength of the current in a nearby conductor.

This phenomenon was discovered by Michael Faraday on August 29, 1831. His experiments showed that the EMF (electromotive force) that appears in a circuit limited by conductors is directly proportional to the rate of change of the flow passing through the area of \u200b\u200bthis circuit.

Important! For the occurrence of EMF, the wire must cross the lines of force. When moving along the lines, there is no EMF.

If the coil in which the EMF occurs is included in the electrical circuit, then a current appears in the winding, which creates its own electromagnetic field in the inductor.

Right hand rule

When a conductor moves in a magnetic field, an EMF is induced in it. Its directionality depends on the direction of wire movement. The method by which the direction of magnetic induction is determined is called the "method right hand».

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Right hand rule

The calculation of the magnitude of the magnetic field is important for the design of electrical machines and transformers.

Video

Let there be a magnetic field in some small area of space, which can be considered homogeneous, that is, in this area the magnetic induction vector is constant, both in magnitude and in direction.
Select a small area ∆S, whose orientation is given by the unit normal vector n(Fig. 445).

rice. 445
Magnetic flux through this pad ΔФ m is defined as the product of the site area and the normal component of the magnetic field induction vector

Where

dot product of vectors B and n;
B n− normal to the site component of the magnetic induction vector.
In an arbitrary magnetic field, the magnetic flux through an arbitrary surface is determined as follows (Fig. 446):

rice. 446
− the surface is divided into small areas ∆S i(which can be considered flat);
− the induction vector is determined B i on that site (which may be considered permanent within the site);
− the sum of flows through all areas into which the surface is divided is calculated

This amount is called flux of the magnetic field induction vector through a given surface (or magnetic flux).
Please note that when calculating the flux, the summation is carried out over the observation points of the field, and not over the sources, as when using the superposition principle. Therefore, the magnetic flux is an integral characteristic of the field, which describes its averaged properties over the entire surface under consideration.
Difficult to find physical meaning magnetic flux, as well as for other fields, this is a useful auxiliary physical quantity. But unlike other fluxes, the magnetic flux is so common in applications that in the SI system it was awarded a "personal" unit of measurement - Weber 2: 1 Weber− magnetic flux of a homogeneous magnetic field of induction 1 T across the square 1 m 2 oriented perpendicular to the magnetic induction vector.
Now let's prove a simple but extremely important theorem about the magnetic flux through a closed surface.
Earlier we established that the forces of any magnetic field are closed, it already follows from this that the magnetic flux through any closed surface zero.

However, we give a more formal proof of this theorem.
First of all, we note that the principle of superposition is valid for a magnetic flux: if a magnetic field is created by several sources, then for any surface the field flux created by a system of current elements is equal to the sum of the field fluxes created by each current element separately. This statement follows directly from the principle of superposition for the induction vector and directly proportional relationship between the magnetic flux and the magnetic induction vector. Therefore, it is sufficient to prove the theorem for the field created by the current element, the induction of which is determined by the Biot-Savarre-Laplace law. Here, the structure of the field, which has axial circular symmetry, is important for us, the value of the modulus of the induction vector is insignificant.
We choose as a closed surface the surface of a bar cut out, as shown in Fig. 447.

rice. 447
The magnetic flux is different from zero only through its two side faces, but these fluxes have opposite signs. Recall that for a closed surface, the outer normal is chosen, therefore, on one of the indicated faces (front), the flow is positive, and on the back, negative. Moreover, the modules of these flows are equal, since the distribution of the field induction vector on these faces is the same. This result does not depend on the position of the considered bar. An arbitrary body can be divided into infinitely small parts, each of which is similar to the considered bar.
Finally, we formulate one more important property flow of any vector field. Let an arbitrary closed surface limit some body (Fig. 448).

rice. 448
Let's split this body into two parts bounded by parts of the original surface Ω 1 and Ω2, and close them with a common interface of the body. The sum of the flows through these two closed surfaces is equal to the flow through the original surface! Indeed, the sum of flows through the boundary (once for one body, another time for another) is equal to zero, since in each case it is necessary to take different, opposite normals (each time external). Similarly, one can prove the statement for an arbitrary partition of the body: if the body is divided into an arbitrary number of parts, then the flow through the surface of the body is equal to the sum of the flows through the surfaces of all parts of the partition of the body. This statement is obvious for fluid flow.
In fact, we have proved that if the flow of a vector field is equal to zero through some surface bounding a small volume, then this flow is equal to zero through any closed surface.
So, for any magnetic field, the magnetic flux theorem is valid: the magnetic flux through any closed surface is equal to zero Ф m = 0.
Previously, we considered flow theorems for the fluid velocity field and the electrostatic field. In these cases, the flow through the closed surface was completely determined by the point sources of the field (fluid sources and sinks, point charges). In the general case, the presence of a nonzero flux through a closed surface indicates the presence of point sources of the field. Hence, the physical content of the magnetic flux theorem is the statement about the absence of magnetic charges.

If you are well versed in this issue and are able to explain and defend your point of view, then you can formulate the magnetic flux theorem like this: “No one has yet found the Dirac monopole.”

It should be specially emphasized that, speaking of the absence of field sources, we mean precisely point sources, similar to electric charges. If we draw an analogy with the field of a moving fluid, electric charges are like points from which fluid flows out (or flows in), increasing or decreasing its amount. The emergence of a magnetic field due to the movement of electric charges is similar to the movement of a body in a liquid, which leads to the appearance of vortices that do not change the total amount of liquid.

Vector fields for which the flow through any closed surface is equal to zero received a beautiful, exotic name − solenoidal. A solenoid is a wire coil through which an electric current can be passed. Such a coil can create strong magnetic fields, so the term solenoidal means "similar to the field of a solenoid", although such fields could be called simpler - "magnetic-like". Finally, such fields are also called eddy, like the velocity field of a fluid that forms all kinds of turbulent eddies in its motion.

The magnetic flux theorem has great importance, it is often used in the proof of various properties of magnetic interactions, we will meet with it repeatedly. For example, the magnetic flux theorem proves that the magnetic field induction vector generated by an element cannot have a radial component, otherwise the flux through a cylindrical coaxial surface with a current element would be nonzero.
Let us now illustrate the application of the magnetic flux theorem to the calculation of the magnetic field induction. Let the magnetic field be created by a ring with a current, which is characterized by a magnetic moment pm. Consider the field near the axis of the ring at a distance z from the center, much larger than the radius of the ring (Fig. 449).

rice. 449
Previously, we obtained a formula for the magnetic field induction on the axis for large distances from the center of the ring

We will not make a big mistake if we assume that the vertical (let the axis of the ring is vertical) component of the field has the same value within a small ring of radius r, whose plane is perpendicular to the axis of the ring. Since the vertical component of the field changes with distance, radial field components must inevitably be present, otherwise the magnetic flux theorem will not hold! It turns out that this theorem and formula (3) are sufficient to find this radial component. Select a thin cylinder with thickness Δz and radius r, whose lower base is at a distance z from the center of the ring, coaxial with the ring, and apply the magnetic flux theorem to the surface of this cylinder. The magnetic flux through the lower base is (note that the induction and normal vectors are opposite here)

where Bz(z) z;
the flow through the top base is

where Bz (z + Δz)− value of the vertical component of the induction vector at height z + z;
flow through side surface(it follows from the axial symmetry that the modulus of the radial component of the induction vector B r on this surface is constant):

According to the proved theorem, the sum of these flows is equal to zero, so the equation

from which we determine the desired value

It remains to use formula (3) for the vertical component of the field and perform the necessary calculations 3

Indeed, a decrease in the vertical component of the field leads to the appearance of horizontal components: a decrease in outflow through the bases leads to a “leakage” through the side surface.
Thus, we have proved the “criminal theorem”: if less flows out through one end of the pipe than is poured into it from the other end, then somewhere they steal through the side surface.

1 It is enough to take the text with the definition of the flow of the tension vector electric field and change the notation (which is done here).
2 Named after the German physicist (member of the St. Petersburg Academy of Sciences) Wilhelm Eduard Weber (1804 - 1891)
3 The most literate can see the derivative of the function (3) in the last fraction and simply calculate it, but we will once again have to use the approximate formula (1 + x) β ≈ 1 + βx.

Right hand or gimlet rule:

The direction of the magnetic field lines and the direction of the current that creates it are interconnected by the well-known rule of the right hand or gimlet, which was introduced by D. Maxwell and is illustrated by the following figures:

Few people know that a gimlet is a tool for drilling holes in a tree. Therefore, it is more understandable to call this rule the rule of a screw, screw or corkscrew. However, grasping the wire as in the figure is sometimes life-threatening!

Magnetic induction B :

Magnetic induction- is the main fundamental characteristic of the magnetic field, similar to the electric field strength vector E . The vector of magnetic induction is always directed tangentially to the magnetic line and shows its direction and strength. The unit of magnetic induction in B = 1 T is the magnetic induction uniform field, in which on a section of the conductor with a length of l\u003d 1 m, with a current strength in it in I\u003d 1 A, the maximum Ampere force acts from the side of the field - F\u003d 1 H. The direction of Ampère's force is determined by the rule of the left hand. In the CGS system, the magnetic induction of the field is measured in gauss (Gs), in the SI system - in teslas (Tl).

Magnetic field strength H:

Another characteristic of the magnetic field is tension, which is analogous to the electric displacement vector D in electrostatics. Determined by the formula:

The magnetic field strength is a vector quantity, it is a quantitative characteristic of the magnetic field and does not depend on magnetic properties environment. In the CGS system, the magnetic field strength is measured in oersteds (Oe), in the SI system - in amperes per meter (A / m).

Magnetic flux F:

Magnetic flux Ф is a scalar physical quantity that characterizes the number of magnetic induction lines penetrating a closed circuit. Consider special case. AT uniform magnetic field, whose induction vector modulus is equal to ∣В ∣, is placed flat closed loop area S. The normal n to the contour plane makes an angle α with the direction of the magnetic induction vector B . The magnetic flux through the surface is the value Ф, determined by the relation:

In the general case, the magnetic flux is defined as the integral of the magnetic induction vector B through the finite surface S.

It is worth noting that the magnetic flux through any closed surface is zero (Gauss's theorem for magnetic fields). This means that the lines of force of the magnetic field do not break anywhere, i.e. the magnetic field has a vortex nature, and also that it is impossible for the existence of magnetic charges that would create a magnetic field in the same way that electric charges create electric field. In SI, the unit of magnetic flux is Weber (Wb), in the CGS system - maxwell (Mks); 1 Wb = 10 8 µs.

Definition of inductance:

Inductance is the coefficient of proportionality between the electric current flowing in any closed circuit and the magnetic flux created by this current through the surface, the edge of which is this circuit.

Otherwise, inductance is the proportionality factor in the self-induction formula.

In the SI system, inductance is measured in henries (H). The circuit has an inductance of one henry if, when the current changes by one ampere per second, EMF self-induction to one volt.

The term "inductance" was proposed by Oliver Heaviside, an English self-taught scientist in 1886. Simply put, inductance is the property of a current-carrying conductor to store energy in a magnetic field, equivalent to capacitance for an electric field. It does not depend on the magnitude of the current, but only on the shape and size of the current-carrying conductor. To increase the inductance, the conductor is wound in coils, the calculation of which is the program

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:

The magnetic flux of a 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 measurement of the flux of magnetic induction in international system units (SI) is 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 a solenoid that has N turns, core length l, area cross section S, the magnetic permeability of the core. The current flowing through the solenoid is I.

Decision

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.
Decision	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"):

If the electric current, as Oersted's experiments showed, creates a magnetic field, then can't the magnetic field in turn induce an electric current in the conductor? Many scientists with the help of experiments tried to find the answer to this question, but Michael Faraday (1791 - 1867) was the first to solve this problem.
In 1831, Faraday discovered that an electric current arises in a closed conducting circuit when the magnetic field changes. This current is called induction current.
Induction current in a coil of metal wire occurs when the magnet is pushed into the coil and when the magnet is pulled out of the coil (Fig. 192),

and also when the current strength changes in the second coil, the magnetic field of which penetrates the first coil (Fig. 193).

The phenomenon of the occurrence of an electric current in a closed conducting circuit with changes in the magnetic field penetrating the circuit is called electromagnetic induction.
The appearance of an electric current in a closed circuit with changes in the magnetic field penetrating the circuit indicates the action of external forces of a non-electrostatic nature in the circuit or the occurrence EMF of induction. Quantitative description of the phenomenon electromagnetic induction is given on the basis of establishing a connection between the induction emf and a physical quantity called magnetic flux.
magnetic flux. For a flat circuit located in a uniform magnetic field (Fig. 194), the magnetic flux F through a surface area S call the value equal to the product of the modulus of the magnetic induction vector and the area S and by the cosine of the angle between the vector and the normal to the surface:

Lenz's rule. Experience shows that the direction of the inductive current in the circuit depends on whether the magnetic flux penetrating the circuit increases or decreases, as well as on the direction of the magnetic field induction vector relative to the circuit. General rule, allowing to determine the direction of the induction current in the circuit, was established in 1833 by E. X. Lenz.
Lenz's rule can be visualized with with the help of a lung aluminum ring (Fig. 195).

Experience shows that when a permanent magnet is introduced, the ring is repelled from it, and when removed, it is attracted to the magnet. The result of the experiments does not depend on the polarity of the magnet.
The repulsion and attraction of a solid ring is explained by the occurrence of an induction current in the ring with changes in the magnetic flux through the ring and the action on induction current magnetic field. Obviously, when the magnet is pushed into the ring, the induction current in it has such a direction that the magnetic field created by this current counteracts the external magnetic field, and when the magnet is pushed out, the induction current in it has such a direction that the induction vector of its magnetic field coincides in direction with the vector external field induction.
General wording Lenz's rules: the induction current arising in a closed circuit has such a direction that the magnetic flux created by it through the area bounded by the circuit tends to compensate for the change in the magnetic flux that causes this current.
The law of electromagnetic induction. Pilot study dependence of the induction emf on the change in the magnetic flux led to the establishment law of electromagnetic induction: The induction emf in a closed loop is proportional to the rate of change of the magnetic flux through the surface bounded by the loop.
In SI, the unit of magnetic flux is chosen such that the coefficient of proportionality between the induction emf and the change in magnetic flux is equal to one. Wherein law of electromagnetic induction is formulated as follows: EMF of induction in a closed loop is equal to the modulus of the rate of change of the magnetic flux through the surface bounded by the loop:

Taking into account the Lenz rule, the law of electromagnetic induction is written as follows:

EMF of induction in the coil. If identical changes in the magnetic flux occur in series-connected circuits, then the induction EMF in them is equal to the sum of the induction EMF in each of the circuits. Therefore, when changing the magnetic flux in the coil, consisting of n identical turns of wire, the total induction emf in n times more EMF induction in a single circuit:

For a uniform magnetic field, on the basis of equation (54.1), it follows that its magnetic induction is 1 T, if the magnetic flux through a 1 m 2 circuit is 1 Wb:

Vortex electric field. The law of electromagnetic induction (54.3) according to known speed changes in the magnetic flux allows you to find the value of the induction EMF in the circuit and at known value electrical resistance loop calculate the current in the loop. However, the physical meaning of the phenomenon of electromagnetic induction remains undisclosed. Let's consider this phenomenon in more detail.

The occurrence of an electric current in a closed circuit indicates that when the magnetic flux penetrating the circuit changes, forces act on free electric charges in the circuit. The wire of the circuit is motionless, free electric charges in it can be considered motionless. Only an electric field can act on stationary electric charges. Therefore, with any change in the magnetic field in the surrounding space, an electric field arises. This electric field sets in motion free electric charges in the circuit, creating an induction electric current. The electric field that occurs when the magnetic field changes is called vortex electric field.

The work of the forces of the vortex electric field on the movement of electric charges is the work of external forces, the source of the induction EMF.

A vortex electric field differs from an electrostatic field in that it is not related to electric charges, its lines of tension are closed lines. The work of the forces of the vortex electric field during the movement of an electric charge along closed line may be different from zero.

EMF of induction in moving conductors. The phenomenon of electromagnetic induction is also observed in cases where the magnetic field does not change in time, but the magnetic flux through the circuit changes due to the movement of the circuit conductors in the magnetic field. In this case, the cause of the induction EMF is not the vortex electric field, but the Lorentz force.