Then the lines of induction of the magnetic field will pass through this circuit. The line of magnetic induction is the magnetic induction at each point of this line. That is, we can say that the lines of magnetic induction are the flow of the induction vector over the space limited and described by these lines. You can say shorter magnetic flux.
In general terms, the concept of "magnetic flux" is introduced in the ninth grade. A more detailed consideration with the derivation of formulas, etc., refers to the senior physics course. So, the magnetic flux is a certain amount of magnetic field induction in any area of space.
Direction and amount of magnetic flux
The magnetic flux has a direction and a quantitative value. In our case, a circuit with current, they say that this circuit permeates a certain magnetic flux. It is clear that the larger the circuit, the greater the magnetic flux will pass through it.
That is, the magnetic flux depends on the area of space through which it passes. If we have a fixed frame of a certain size, penetrated by a constant magnetic field, then the magnetic flux passing through this frame will be constant.
If we increase the strength of the magnetic field, then the magnetic induction will increase accordingly. The magnitude of the magnetic flux will also increase, and in proportion to the increased magnitude of the induction. That is, the magnetic flux depends on the magnitude of the magnetic field induction and the area of the penetrating surface.
Magnetic flux and frame - consider an example
Consider the option when our frame is located perpendicular to the magnetic flux. The area bounded by this frame will be maximum in relation to the magnetic flux passing through it. Therefore, the value of the flux will be maximum for a given value of the magnetic field induction.
If we begin to rotate the frame relative to the direction of the magnetic flux, then the area through which the magnetic flux can pass will decrease, therefore, the magnitude of the magnetic flux through this frame will decrease. Moreover, it will decrease down to zero when the frame becomes parallel to the lines of magnetic induction.
The magnetic flux will, as it were, slide past the frame, it will not penetrate it. In this case, the effect of the magnetic field on the frame with current will be zero. Thus, we can deduce the following dependency:
The magnetic flux penetrating the circuit area changes when the module of the magnetic induction vector B, the circuit area S changes, and when the circuit rotates, that is, when its orientation to the magnetic field induction lines changes.
Among the many definitions and concepts associated with a magnetic field, one should highlight the magnetic flux, which has a certain direction. This property is widely used in electronics and electrical engineering, in the design of instruments and devices, as well as in the calculation of various circuits.
The concept of magnetic flux
First of all, it is necessary to establish exactly what is called magnetic flux. This value should be considered in combination with a uniform magnetic field. It is homogeneous at every point of the designated space. A certain surface, which has some fixed area, denoted by the symbol S, falls under the action of a magnetic field. The field lines act on this surface and cross it.
Thus, the magnetic flux Ф, crossing the surface with area S, consists of a certain number of lines coinciding with the vector B and passing through this surface.
This parameter can be found and displayed in the form of the formula Ф = BS cos α, in which α is the angle between the normal direction to the surface S and the magnetic induction vector B. Based on this formula, one can determine the magnetic flux with a maximum value at which cos α = 1 , and the position of the vector B will become parallel to the normal perpendicular to the surface S. Conversely, the magnetic flux will be minimal if the vector B is located perpendicular to the normal.
In this version, the vector lines simply slide along the plane and do not cross it. That is, the flux is taken into account only along the lines of the magnetic induction vector crossing a specific surface.
To find this value, weber or volt-seconds are used (1 Wb \u003d 1 V x 1 s). This parameter can be measured in other units. The smaller value is the maxwell, which is 1 Wb = 10 8 µs or 1 µs = 10 -8 Wb.
Magnetic field energy and magnetic induction flux
If an electric current is passed through a conductor, then a magnetic field is formed around it, which has energy. Its origin is associated with the electric power of the current source, which is partially consumed to overcome the EMF of self-induction that occurs in the circuit. This is the so-called self-energy of the current, due to which it is formed. That is, the energies of the field and current will be equal to each other.
The value of the self-energy of the current is expressed by the formula W \u003d (L x I 2) / 2. This definition is considered equal to the work that is done by a current source that overcomes the inductance, that is, the self-induction EMF and creates a current in the electrical circuit. When the current stops acting, the energy of the magnetic field does not disappear without a trace, but is released, for example, in the form of an arc or spark.
The magnetic flux that occurs in the field is also known as the flux of magnetic induction with a positive or negative value, the direction of which is conventionally indicated by a vector. As a rule, this flow passes through a circuit through which an electric current flows. With a positive direction of the normal relative to the contour, the direction of current movement is a value determined in accordance with . In this case, the magnetic flux created by the circuit with electric current, and passing through this circuit, will always have a value greater than zero. Practical measurements also point to this.
The magnetic flux is usually measured in units established by the international SI system. This is the already known Weber, which is the magnitude of the flow passing through a plane with an area of 1 m2. This surface is placed perpendicular to the magnetic field lines with a uniform structure.
This concept is well described by the Gauss theorem. It reflects the absence of magnetic charges, so the induction lines are always represented as closed or going to infinity without beginning or end. That is, the magnetic flux passing through any kind of closed surfaces is always zero.
Electric dipole moment
Electric charge
electrical induction
Electric field
electrostatic potential
| magnetostatics |
|---|
| Biot-Savart-Laplace law Ampère's law Magnetic moment A magnetic field magnetic flux Magnetic induction |
| Electrodynamics |
|---|
| vector potential Dipole Potentials of Lienar - Wiechert Lorentz force Bias current Unipolar induction Maxwell's equations Electricity Electromotive force Electromagnetic induction Electromagnetic radiation Electromagnetic field |
| Electrical circuit |
|---|
| Ohm's law Kirchhoff's laws Inductance radio waveguide Resonator Electric capacity electrical conductivity Electrical resistance Electrical impedance |
| Notable scientists |
|---|
| Henry Cavendish Michael Faraday Nikola Tesla André-Marie Ampère Gustav Robert Kirchhoff James Clerk (Clark) Maxwell Henry Rudolf Hertz Albert Abraham Michelson Robert Andrews Milliken |
magnetic flux- physical quantity equal to the product of the modulus of the magnetic induction vector to the area S and the cosine of the angle α between vectors and normal . Flow as an integral of the magnetic induction vector through the end surface S is defined via the integral over the surface:
{{{1}}}In this case, the vector element d S surface area S defined as
{{{1}}}Magnetic flux quantization
The values of the magnetic flux Φ passing through
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An excerpt characterizing the Magnetic flux
- C "est bien, mais ne demenagez pas de chez le prince Basile. Il est bon d" avoir un ami comme le prince, she said, smiling at Prince Vasily. - J "en sais quelque chose. N" est ce pas? [That's good, but don't move away from Prince Vasily. It's good to have such a friend. I know something about it. Isn't it?] And you're still so young. You need advice. You are not angry with me that I use the rights of old women. - She fell silent, as women are always silent, waiting for something after they say about their years. - If you marry, then another matter. And she put them together in one look. Pierre did not look at Helen, and she at him. But she was still terribly close to him. He mumbled something and blushed.Returning home, Pierre could not sleep for a long time, thinking about what had happened to him. What happened to him? Nothing. He only realized that the woman he knew as a child, about whom he absentmindedly said: “Yes, good,” when he was told that Helen was beautiful, he realized that this woman could belong to him.
“But she is stupid, I myself said she was stupid,” he thought. - There is something nasty in the feeling that she aroused in me, something forbidden. I was told that her brother Anatole was in love with her, and she was in love with him, that there was a whole story, and that Anatole was expelled from this. Her brother is Ippolit... Her father is Prince Vasily... This is not good, he thought; and at the same time as he was reasoning like this (these reasonings were still unfinished), he forced himself to smile and realized that another series of reasonings had surfaced because of the first ones, that at the same time he was thinking about her insignificance and dreaming about how she would be his wife, how she could love him, how she could be completely different, and how everything he thought and heard about her could be untrue. And he again saw her not as some kind of daughter of Prince Vasily, but saw her whole body, only covered with a gray dress. “But no, why didn’t this thought occur to me before?” And again he told himself that it was impossible; that something nasty, unnatural, as it seemed to him, dishonest would be in this marriage. He remembered her former words, looks, and the words and looks of those who had seen them together. He remembered the words and looks of Anna Pavlovna when she told him about the house, remembered thousands of such hints from Prince Vasily and others, and he was horrified that he had not bound himself in any way in the performance of such a thing, which, obviously, was not good. and which he must not do. But at the same time as he was expressing this decision to himself, from the other side of his soul her image surfaced with all its feminine beauty.
In November 1805, Prince Vasily had to go to four provinces for an audit. He arranged this appointment for himself in order to visit his ruined estates at the same time, and taking with him (at the location of his regiment) his son Anatole, together with him to call on Prince Nikolai Andreevich Bolkonsky in order to marry his son to the daughter of this rich old man. But before leaving and these new affairs, Prince Vasily had to resolve matters with Pierre, who, it is true, had spent whole days at home, that is, with Prince Vasily, with whom he lived, he was ridiculous, agitated and stupid (as he should being in love) in Helen's presence, but still not proposing.
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.
It is difficult to find the physical meaning of the magnetic flux, as for other fields it 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 is 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 of the 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, we 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. Consequently, 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 is of great importance, it is often used to prove 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 field component changes with distance, radial field components must inevitably be present, otherwise the 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 the 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 flux of the electric field strength vector 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 have to use the approximate formula (1 + x) β ≈ 1 + βx once again.
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 magnetic induction vector 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 of a homogeneous field, in which 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 the magnetic properties of the medium. 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 loop. 
Let's consider a 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 an 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). A circuit has an inductance of one henry if, when the current changes by one ampere per second, a self-induction emf of one volt occurs at the circuit terminals.
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
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