What is the flux of magnetic induction through the circuit. Magnetic flux and flux linkage

In order to understand the meaning of the concept of “magnetic flux”, which is new for us, we will analyze in detail several experiments with EMF guidance, paying attention to the quantitative side of the observations made.

In our experiments, we will use the setup shown in Fig. 2.24.

It consists of a large multi-turn coil wound, say, on a tube of thick glued cardboard. The coil is powered from the battery through a switch and an adjusting rheostat. The magnitude of the current established in the coil can be judged by an ammeter (not shown in Fig. 2.24).

Inside the large coil, another small coil can be installed, the ends of which are connected to a magnetoelectric device - a galvanometer.

For the sake of illustration, part of the coil is shown cut out - this allows you to see the location of the small coil.

When the switch is closed or opened in a small coil, an EMF is induced and the galvanometer needle points to a short time dropped from the zero position.

According to the deviation, one can judge in which case the induced emf is greater, in which it is less.

Rice. 2.24. A device on which you can study the induction of EMF by a changing magnetic field

Noticing the number of divisions to which the arrow is thrown, one can quantitatively compare the effect produced by the induced EMF.

First observation. Inserting a small one inside the large coil, we fix it and for now we will not change anything in their location.

Turn on the switch and, changing the resistance of the rheostat connected after the battery, set certain value current, for example

Let us now turn off the switch, observing the galvanometer. Let its offset n be equal to 5 divisions to the right:

When the current is 1 A.

Turn on the switch again and, by changing the resistance, increase the current of the large coil to 4 A.

Let's let the galvanometer calm down, and turn off the switch again, watching the galvanometer.

If its rejection was 5 divisions when the current was turned off at 1 A, then now, when turning off 4 A, we note that the rejection increased by 4 times:

When the 4A current is turned off.

Continuing such observations, it is easy to conclude that the rejection of the galvanometer, and hence the induced EMF, increase in proportion to the growth of the current to be switched off.

But we know that a change in current causes a change magnetic field(of his induction), so the correct conclusion from our observation is:

the induced emf is proportional to the rate of change of magnetic induction.

More detailed observations confirm the correctness of this conclusion.

Second observation. Let us continue to observe the rejection of the galvanometer by turning off the same current, say, 1-4 A. But we will change the number of turns N of a small coil, leaving its location and dimensions unchanged.

Let us assume that the rejection of the galvanometer

was observed at (100 turns on a small coil).

How will the offset of the galvanometer change if the number of turns is doubled?

Experience shows that

This is exactly what was to be expected.

In fact, all turns of a small coil are under the same influence of a magnetic field, and the same EMF must be induced in each turn.

Let us denote the EMF of one turn by the letter E, then the EMF of 100 turns, connected in series one after the other, should be 100 times greater:

At 200 turns

For any other number of turns

If the emf increases in proportion to the number of turns, then it goes without saying that the rejection of the galvanometer must also be proportional to the number of turns.

This is what experience shows. So,

the induced emf is proportional to the number of turns.

We emphasize once again that the dimensions of the small coil and its arrangement remained unchanged during our experiment. It goes without saying that the experiment was carried out in the same large coil with the same current turned off.

Third observation. Having done several experiments with the same small coil with the current turned on unchanged, it is easy to verify that the magnitude of the induced EMF depends on how the small coil is located.

To observe the dependence of the induced EMF on the position of a small coil, we will improve our installation somewhat (Fig. 2.25).

To the outward end of the axis of a small coil, we will attach an index arrow and a circle with division (like

Rice. 2.25. A device for turning a small coil fixed on a rod passed through the walls of a large coil. The rod is connected to the index arrow. The position of the arrow on the half-ring with divisions shows how the small coil of those that can be found on radios is located).

By turning the rod, we can now judge by the position of the index arrow the position that the small coil occupies inside the large one.

Observations show that

the greatest EMF is induced when the axis of the small coil coincides with the direction of the magnetic field,

in other words, when the axes of the large and small coils are parallel.

Rice. 2.26. To the conclusion of the concept of "magnetic flux". The magnetic field is depicted by lines drawn at the rate of two lines per 1 cm2: a - a coil with an area of 2 cm2 is located perpendicular to the direction of the field. A magnetic flux is coupled to each turn of the coil. This flux is depicted by four lines crossing the coil; b - a coil with an area of 4 cm2 is located perpendicular to the direction of the field. A magnetic flux is coupled to each turn of the coil. This flux is depicted by eight lines crossing the coil; c - a coil with an area of 4 cm2 is located obliquely. magnetic flux, linked to each of its coils, is depicted by four lines. It is equal since each line depicts, as can be seen from Fig. 2.26, a and b, flow c. The flux coupled to the coil is reduced due to its inclination.

This arrangement of the small coil is shown in Fig. 2.26, a and b. As the coil turns, the EMF induced in it will be less and less.

Finally, if the plane of the small coil becomes parallel to the lines, the field, no EMF will be induced in it. The question may arise, what will happen with the further rotation of the small coil?

If we turn the coil more than 90° (relative to the initial position), then the sign of the induced emf will change. The field lines will enter the coil from the other side.

Fourth observation. It is important to make one more final observation.

Let's choose a certain position in which we will put a small coil.

Let us agree, for example, to always put it in such a position that the induced EMF is as large as possible (of course, for a given number of turns and given value switched off current). We will make several small coils of different diameters, but with the same number turns.

We will put these coils in the same position and, turning off the current, we will observe the rejection of the galvanometer.

Experience will show us that

induced emf is proportional to the area cross section coils.

magnetic flux. All observations allow us to conclude that

the induced emf is always proportional to the change in magnetic flux.

But what is magnetic flux?

First, we will talk about the magnetic flux through a flat area S, forming a right angle with the direction of the magnetic field. In this case, the magnetic flux is equal to the product of the area and the induction, or

here S is the area of our site, m2;; B - induction, T; Ф - magnetic flux, Wb.

The unit of flow is the weber.

Depicting the magnetic field through lines, we can say that the magnetic flux is proportional to the number of lines penetrating the area.

If the field lines are drawn in such a way that their number on a perpendicularly set plane is equal to the field induction B, then the flow is equal to the number of such lines.

On fig. 2.26 the magnetic lule in is shown by lines drawn on the basis of two lines per line. Each line, therefore, corresponds to a magnetic flux of magnitude

Now, in order to determine the magnitude of the magnetic flux, it is enough to simply count the number of lines penetrating the area and multiply this number by

In the case of Fig. 2.26, and the magnetic flux through an area of 2 cm2, perpendicular to the direction of the field,

On fig. 2.26, and this area is pierced by four magnetic lines. In the case of Fig. 2.26, b magnetic flux through a transverse platform of 4 cm2 at an induction of 0.2 T

and we see that the platform is pierced by eight magnetic lines.

Magnetic flux coupled to a coil. Speaking of the induced emf, we need to keep in mind the flux coupled to the coil.

A flow coupled to a coil is a flow penetrating the surface bounded by the coil.

On fig. 2.26 the flow coupled with each turn of the coil, in the case of fig. 2.26, a is equal to a in the case of fig. 2.26, b flow is

If the platform is not perpendicular, but inclined to magnetic lines, then it is no longer possible to determine the flow simply by the product of the area and the induction. The flow in this case is defined as the product of induction and the projection area of our site. It's about about the projection onto a plane perpendicular to the lines of the field, or, as it were, about the shadow cast by the site (Fig. 2.27).

However, for any shape of the pad, the flow is still proportional to the number of lines passing through it, or equal to the number of unit lines penetrating the pad.

Rice. 2.27. To the conclusion of the site projection. Carrying out the experiments in more detail and combining our third and fourth observations, one could draw the following conclusion; the induced emf is proportional to the area of the shadow cast by our little coil on a plane perpendicular to the field lines, if it were illuminated by rays of light parallel to the field lines. Such a shadow is called a projection.

So, in fig. 2.26, in a flow through a platform of 4 cm2 at an induction of 0.2 T, it is equal to everything (lines with a price of ). The representation of the magnetic field by lines is very helpful in determining the flux.

If each of the N turns of the coil is coupled with a flux Ф, we can call the product NF the total flux linkage of the coil. The concept of flux linkage can be used especially conveniently when different threads are linked to different coils. In this case, the total flux linkage is the sum of the fluxes linked to each of the turns.

A few notes about the word "flow". Why are we talking about flow? Is the idea of some kind of flow of something magnetic connected with this word? In fact, when we say "electric current", we imagine the movement (flow) of electric charges. Is it the same in the case of magnetic flux?

No, when we say "magnetic flux" we mean only a certain measure of the magnetic field (the product of the field strength and the area), similar to the measure used by engineers and scientists who study the movement of fluids. When water moves, they call it the flow of the product of the water velocity and the area of the transversely located area (the flow of water in the pipe is equal to its velocity and the cross-sectional area of \u200b\u200bthe pipe).

Of course, the magnetic field itself, which is one of the types of matter, is also associated with a special form of motion. We still do not have sufficiently clear ideas and knowledge about the nature of this movement, although modern scientists know a lot about the properties of the magnetic field: the magnetic field is associated with the existence of a special form of energy, its main measure is induction, another very important measure is the 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.

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 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 EMF self-induction to create 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.

Decision

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.

Decision

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 emf of induction and induction current 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 about , the quantity is dimensionless.

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

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

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

where AT " - 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 about , the expression is true: AT " = c about AT 0 (2.6)

Divide by m 0 expression (2.6):

AT " /m about = c about AT 0 /m 0

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

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

I=c about 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.).

Using lines of force, one can not only show the direction of the magnetic field, but also characterize the magnitude of its induction.

We agreed to draw lines of force in such a way that through 1 cm² of the area, perpendicular to the induction vector at a certain point, the number of lines equal to the field induction at this point passed.

In the place where the field induction is greater, the lines of force will be thicker. And, conversely, where the field induction is less, the lines of force are rarer.

A magnetic field with the same induction at all points is called a uniform field. Graphically, a uniform magnetic field is represented by lines of force, which are equally spaced from each other.

An example homogeneous field is the field inside the long solenoid, as well as the field between closely spaced parallel flat pole pieces of the electromagnet.

The product of the induction of a magnetic field penetrating a given circuit by the area of \u200b\u200bthe circuit is called the magnetic flux of magnetic induction, or simply magnetic flux.

The English physicist Faraday gave him a definition and studied his properties. He discovered that this concept allows a deeper consideration of the unified nature of magnetic and electrical phenomena.

Denoting the magnetic flux with the letter F, the area of the circuit S and the angle between the direction of the induction vector B and the normal n to the area of the circuit α, we can write the following equality:

Ф = В S cos α.

Magnetic flux is a scalar quantity.

Since the density of the lines of force of an arbitrary magnetic field is equal to its induction, the magnetic flux is equal to the entire number of lines of force that permeate this circuit.

With a change in the field, the magnetic flux that permeates the circuit also changes: when the field is strengthened, it increases, and when the field is weakened, it decreases.

The unit of magnetic flux in is taken to be the flux that permeates an area of 1 m², located in a magnetic uniform field, with an induction of 1 Wb / m², and located perpendicular to the induction vector. Such a unit is called a weber:

1 Wb \u003d 1 Wb / m² ˖ 1 m².

The changing magnetic flux generates an electric field with closed lines of force (vortex electric field). Such a field manifests itself in the conductor as the action of extraneous forces. This phenomenon is called electromagnetic induction, and the electromotive force that arises in this case is called the induction EMF.

In addition, it should be noted that the magnetic flux makes it possible to characterize the entire magnet as a whole (or any other sources of the magnetic field). Therefore, if it makes it possible to characterize its action at any single point, then the magnetic flux is entirely. That is, we can say that this is the second most important And, therefore, if magnetic induction acts as a force characteristic of a magnetic field, then magnetic flux is its energy characteristic.

Returning to the experiments, we can also say that each coil coil can be imagined as a single closed coil. The same circuit through which the magnetic flux of the magnetic induction vector will pass. In this case, an inductive electric current will be noted. Thus, it is under the influence of a magnetic flux that an electric field is formed in a closed conductor. And then this electric field forms an electric current.

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 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 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. So, for example, the magnetic flux theorem proves that the magnetic field induction vector created 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 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 once again have to use the approximate formula (1 + x) β ≈ 1 + βx.

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 quantization

The values of the magnetic flux Φ passing through

Write a review on the article "Magnetic flux"

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 found himself smiling and realizing 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 settle 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.