The magnetic induction of the field created by an infinitely long straight conductor carrying current is

If you bring the magnetic needle close, it will tend to become perpendicular to the plane passing through the axis of the conductor and the center of rotation of the needle. This indicates that special forces act on the arrow, which are called magnetic forces. In addition to the effect on the magnetic needle, the magnetic field affects moving charged particles and current-carrying conductors located in the magnetic field. In conductors moving in a magnetic field, or in stationary conductors located in an alternating magnetic field, an inductive electromotive force (emf) arises.

A magnetic field

In accordance with the above, we can give the following definition of a magnetic field.

A magnetic field is one of the two sides of the electromagnetic field, excited by the electric charges of moving particles and changes in the electric field and characterized by a force effect on moving infected particles, and therefore on electric currents.

If you pass a thick conductor through cardboard and pass an electric current through it, then the steel filings poured onto the cardboard will be located around the conductor in concentric circles, which in this case are the so-called magnetic induction lines (Figure 1). We can move the cardboard up or down the conductor, but the location of the steel filings will not change. Consequently, a magnetic field arises around the conductor along its entire length.

If you place small magnetic arrows on the cardboard, then by changing the direction of the current in the conductor, you can see that the magnetic arrows will rotate (Figure 2). This shows that the direction of magnetic induction lines changes with the direction of current in the conductor.

Magnetic induction lines around a current-carrying conductor have the following properties: 1) magnetic induction lines of a straight conductor have the shape of concentric circles; 2) the closer to the conductor, the denser the magnetic induction lines are located; 3) magnetic induction (field intensity) depends on the magnitude of the current in the conductor; 4) the direction of magnetic induction lines depends on the direction of the current in the conductor.

To show the direction of the current in the conductor shown in section, a symbol has been adopted, which we will use in the future. If you mentally place an arrow in the conductor in the direction of the current (Figure 3), then in the conductor in which the current is directed away from us, we will see the tail of the arrow’s feathers (a cross); if the current is directed towards us, we will see the tip of an arrow (point).

Figure 3. Symbol for the direction of current in conductors

The gimlet rule allows you to determine the direction of magnetic induction lines around a current-carrying conductor. If a gimlet (corkscrew) with a right-hand thread moves forward in the direction of the current, then the direction of rotation of the handle will coincide with the direction of the magnetic induction lines around the conductor (Figure 4).

A magnetic needle introduced into the magnetic field of a current-carrying conductor is located along the magnetic induction lines. Therefore, to determine its location, you can also use the “gimlet rule” (Figure 5). The magnetic field is one of the most important manifestations of electric current and cannot be obtained independently and separately from the current.


Figure 4. Determining the direction of magnetic induction lines around a current-carrying conductor using the “gimlet rule”	Figure 5. Determining the direction of deviation of a magnetic needle brought to a conductor with current, according to the “gimlet rule”

Magnetic induction

A magnetic field is characterized by a magnetic induction vector, which therefore has a certain magnitude and a certain direction in space.

A quantitative expression for magnetic induction as a result of generalization of experimental data was established by Biot and Savart (Figure 6). Measuring the magnetic fields of electric currents of various sizes and shapes by the deflection of the magnetic needle, both scientists came to the conclusion that every current element creates a magnetic field at some distance from itself, the magnetic induction of which is Δ B is directly proportional to the length Δ l this element, the magnitude of the flowing current I, the sine of the angle α between the direction of the current and the radius vector connecting the field point of interest to us with a given current element, and is inversely proportional to the square of the length of this radius vector r:

Where K– coefficient depending on the magnetic properties of the medium and on the chosen system of units.

In the absolute practical rationalized system of units of ICSA

where µ 0 – magnetic permeability of vacuum or magnetic constant in the MCSA system:

µ 0 = 4 × π × 10 -7 (henry/meter);

Henry (gn) – unit of inductance; 1 gn = 1 ohm × sec.

µ – relative magnetic permeability– a dimensionless coefficient showing how many times the magnetic permeability of a given material is greater than the magnetic permeability of vacuum.

The dimension of magnetic induction can be found using the formula

Volt-second is also called Weber (wb):

In practice, there is a smaller unit of magnetic induction - gauss (gs):

Biot-Savart's law allows us to calculate the magnetic induction of an infinitely long straight conductor:

Where A– the distance from the conductor to the point where the magnetic induction is determined.

Magnetic field strength

The ratio of magnetic induction to the product of magnetic permeabilities µ × µ 0 is called magnetic field strength and is designated by the letter H:

B = H × µ × µ 0 .

The last equation relates two magnetic quantities: induction and magnetic field strength.

Let's find the dimension H:

Sometimes another unit of measurement of magnetic field strength is used - Oersted (er):

1 er = 79,6 A/m ≈ 80 A/m ≈ 0,8 A/cm .

Magnetic field strength H, like magnetic induction B, is a vector quantity.

A line tangent to each point of which coincides with the direction of the magnetic induction vector is called magnetic induction line or magnetic induction line.

Magnetic flux

The product of magnetic induction by the area perpendicular to the field direction (magnetic induction vector) is called flux of the magnetic induction vector or simply magnetic flux and is denoted by the letter F:

F = B × S .

Magnetic flux dimension:

that is, magnetic flux is measured in volt-seconds or webers.

The smaller unit of magnetic flux is Maxwell (mks):

1 wb = 108 mks.
1mks = 1 gs× 1 cm 2.

Video 1. Ampere's hypothesis

Video 2. Magnetism and electromagnetism

Good day to all. In the last article I talked about the magnetic field and dwelled a little on its parameters. This article continues the topic of the magnetic field and is devoted to such a parameter as magnetic induction. To simplify the topic, I will talk about the magnetic field in a vacuum, since different substances have different magnetic properties, and as a result, it is necessary to take their properties into account.

Biot–Savart–Laplace law

As a result of studying the magnetic fields created by electric current, researchers came to the following conclusions:

magnetic induction created by electric current is proportional to the strength of the current;
magnetic induction depends on the shape and size of the conductor through which the electric current flows;
magnetic induction at any point in the magnetic field depends on the location of this point in relation to the current-carrying conductor.

The French scientists Biot and Savard, who came to such conclusions, turned to the great mathematician P. Laplace to generalize and derive the basic law of magnetic induction. He hypothesized that the induction at any point of the magnetic field created by a current-carrying conductor can be represented as the sum of the magnetic inductions of elementary magnetic fields that are created by an elementary section of a current-carrying conductor. This hypothesis became the law of magnetic induction, called Biot-Savart-Laplace law. To consider this law, let us depict a current-carrying conductor and the magnetic induction it creates

Magnetic induction dB created by an elementary section of a conductor dl.

Then magnetic induction dB elementary magnetic field that is created by a section of a conductor dl, with current I at any point R will be determined by the following expression

where I is the current flowing through the conductor,

r is the radius vector drawn from the conductor element to the magnetic field point,

dl is the minimum conductor element that creates induction dB,

k – proportionality coefficient, depending on the reference system, in SI k = μ 0 /(4π)

Because is a vector product, then the final expression for the elementary magnetic induction will look like this

Thus, this expression allows us to find the magnetic induction of the magnetic field, which is created by a conductor with a current of arbitrary shape and size by integrating the right side of the expression

where the symbol l indicates that integration occurs along the entire length of the conductor.

Magnetic induction of a straight conductor

As you know, the simplest magnetic field creates a straight conductor through which electric current flows. As I already said in the previous article, the lines of force of a given magnetic field are concentric circles located around the conductor.

To determine magnetic induction IN straight wire at a point R Let us introduce some notation. Since the point R is at a distance b from the wire, then the distance from any point on the wire to the point R is defined as r = b/sinα. Then the shortest length of the conductor dl can be calculated from the following expression

As a result, the Biot–Savart–Laplace law for a straight wire of infinite length will have the form

where I is the current flowing through the wire,

b is the distance from the center of the wire to the point at which the magnetic induction is calculated.

Now we simply integrate the resulting expression over dα ranging from 0 to π.

Thus, the final expression for the magnetic induction of a straight wire of infinite length will have the form

I – current flowing through the wire,

b is the distance from the center of the conductor to the point at which the induction is measured.

Magnetic induction of the ring

The induction of a straight wire has a small value and decreases with distance from the conductor, therefore it is practically not used in practical devices. The most widely used magnetic fields are those created by a wire wound around a frame. Therefore, such fields are called magnetic fields of circular current. The simplest such magnetic field is possessed by an electric current flowing through a conductor, which has the shape of a circle of radius R.

In this case, two cases are of practical interest: the magnetic field at the center of the circle and the magnetic field at point P, which lies on the axis of the circle. Let's consider the first case.

In this case, each current element dl creates an elementary magnetic induction dB in the center of the circle, which is perpendicular to the contour plane, then the Biot-Savart-Laplace law will have the form

All that remains is to integrate the resulting expression over the entire length of the circle

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

I – current strength in the conductor,

R is the radius of the circle into which the conductor is rolled.

Let's consider the second case, when the point at which the magnetic induction is calculated lies on the straight line X, which is perpendicular to the plane limited by the circular current.

In this case, induction at the point R will be the sum of elementary inductions dB X, which in turn represents a projection onto the axis X elementary induction dB

Applying the Biot-Savart-Laplace law, we calculate the value of magnetic induction

Now let’s integrate this expression over the entire length of the circle

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

I – current strength in the conductor,

R is the radius of the circle into which the conductor is rolled,

x is the distance from the point at which the magnetic induction is calculated to the center of the circle.

As can be seen from the formula for x = 0, the resulting expression transforms into the formula for magnetic induction at the center of the circular current.

Circulation of the magnetic induction vector

To calculate the magnetic induction of simple magnetic fields, the Biot-Savart-Laplace law is sufficient. However, with more complex magnetic fields, for example, the magnetic field of a solenoid or toroid, the number of calculations and the cumbersomeness of the formulas will increase significantly. To simplify calculations, the concept of circulation of the magnetic induction vector is introduced.

Let's imagine some contour l, which is perpendicular to the current I. At any point R of this circuit, magnetic induction IN directed tangentially to this contour. Then the product of vectors dl And IN is described by the following expression

Since the angle dφ small enough, then the vectors dl B defined as arc length

Thus, knowing the magnetic induction of a straight conductor at a given point, we can derive an expression for the circulation of the magnetic induction vector

Now all that remains is to integrate the resulting expression over the entire length of the contour

In our case, the magnetic induction vector circulates around one current, but in the case of several currents, the expression for the circulation of magnetic induction turns into the law of total current, which states:

The circulation of the magnetic induction vector in a closed loop is proportional to the algebraic sum of the currents that the given loop covers.

Magnetic field of solenoid and toroid

Using the law of total current and circulation of the magnetic induction vector, it is quite easy to determine the magnetic induction of such complex magnetic fields as those of a solenoid and a toroid.

A solenoid is a cylindrical coil that consists of many turns of conductor wound turn to turn on a cylindrical frame. The magnetic field of a solenoid actually consists of multiple magnetic fields of a circular current with a common axis perpendicular to the plane of each circular current.

Let's use the circulation of the magnetic induction vector and imagine the circulation along a rectangular contour 1-2-3-4 . Then the circulation of the magnetic induction vector for a given circuit will have the form

Since in the areas 2-3 And 4-1 the magnetic induction vector is perpendicular to the circuit, then the circulation is zero. Location on 3-4 , which is significantly removed from the solenoid, then it can also be ignored. Then, taking into account the law of total current, the magnetic induction in a solenoid of sufficiently large length will have the form

where n is the number of turns of the solenoid conductor per unit length,

I – current flowing through the solenoid.

A toroid is formed by winding a conductor around a ring frame. This design is equivalent to a system of many identical circular currents, the centers of which are located on a circle.

As an example, consider a toroid of radius R, on which it is wound N turns of wire. Around each turn of the wire we take a radius contour r, the center of this contour coincides with the center of the toroid. Since the magnetic induction vector B is directed tangentially to the contour at each point of the contour, then the circulation of the magnetic induction vector will have the form

where r is the radius of the magnetic induction loop.

The circuit passing inside the toroid covers N turns of wire with current I, then the law of the total current for the toroid will have the form

where n is the number of turns of the conductor per unit length,

r – radius of the magnetic induction loop,

R is the radius of the toroid.

Thus, using the law of total current and the circulation of the magnetic induction vector, it is possible to calculate an arbitrarily complex magnetic field. However, the law of total current gives correct results only in a vacuum. When calculating magnetic induction in a substance, it is necessary to take into account the so-called molecular currents. This will be discussed in the next article.

Theory is good, but without practical application it is just words.

Let along the axis OZ There is an infinitely long conductor through which a current flows with force . What is current strength?
,
- charge that crosses the surface S in time
. The system has axial symmetry. If we introduce cylindrical coordinates r,  , z, then cylindrical symmetry means that
and besides,
, when displaced along the axis OZ, we see the same thing. This is the source. The magnetic field must be such that these conditions are satisfied
And
. This means this: magnetic field lines are circles lying in a plane orthogonal to the conductor. This immediately allows you to find the magnetic field.

P This is our guide.

Here is the orthogonal plane,

here is the radius circle r,

I'll take a tangent vector here, a vector directed along  , tangent vector to the circle.

Then,
,
Where
.

For a closed contour, select a circle of radius r= const. We then write, the sum of the lengths along the entire circle (and the integral is nothing more than the sum) is the circumference., where  is the current strength in the conductor. On the right is a charge that crosses the surface per unit time. Hence the moral:
. This means that a straight conductor creates a magnetic field with lines of force in the form of circles surrounding the conductor, and this value IN decreases as we move away from the conductor, well, and tends to infinity if we approach the conductor, when the circuit goes inside the conductor.

E that result is only for the case where the circuit is carrying current. It is clear that an infinite conductor is unrealizable. The length of a conductor is an observable quantity, and no observable quantity can take on infinite values, not with a ruler that would allow one to measure an infinite length. This is an unrealizable thing, then what is the use of this formula? The message is simple. For any conductor, the following will be true: the magnetic field lines are sufficiently close to the conductor - these are closed circles enveloping the conductor, and at a distance
(R– radius of curvature of the conductor), this formula will be valid.

A magnetic field created by an arbitrary current-carrying conductor.

Bio-Savart's Law.

P Let us assume that we have an arbitrary conductor carrying current, and we are interested in the magnetic field created by a piece of this conductor at a given point. How, by the way, in electrostatics did we find the electric field created by some kind of charge distribution? The distribution was divided into small elements and the field from each element was calculated at each point (according to Coulomb’s law) and summed up. The same program is here. The structure of a magnetic field is more complex than an electrostatic one; by the way, it is not potential; a closed magnetic field cannot be represented as a gradient of a scalar function; it has a different structure, but the idea is the same. We break the conductor into small elements. Here I took a small element
, the position of this element is determined by the radius vector , and the observation point is specified by the radius vector . It is stated that this element of the conductor will create induction at this point according to this recipe:
. Where does this recipe come from? It was found experimentally at one time; by the way, it’s hard for me to imagine how it was possible to experimentally find such a rather complex formula with a vector product. This is actually a consequence of Maxwell's fourth equation
. Then the field created by the entire conductor:
, or, we can now write the integral:
. It is clear that calculating such an integral for an arbitrary conductor is not a very pleasant task, but in the form of a sum this is a normal task for a computer.

Example. Magnetic field of a circular coil with current.

P be in plane YZ There is a wire coil of radius R through which a current of force  flows. We are interested in the magnetic field that creates the current. The lines of force near the turn are:

The general picture of the lines of force is also visible ( Fig.7.10).

P about the idea, we would be interested in the field
, but in elementary functions it is impossible to specify the field of this turn. It can only be found on the axis of symmetry. We are looking for a field at points ( X,0,0).

Vector direction determined by the vector product
. Vector has two components:
And . When we start summing these vectors, all the perpendicular components add up to zero.
. And now we write:
,
=, a
.
, and finally 1) ,
.

We got the following result:

And now, as a check, the field in the center of the turn is equal to:
.

Long solenoid field.

A solenoid is a coil on which a conductor is wound.

M the magnetic field from the turns adds up, and it is not difficult to guess that the structure of the field lines is as follows: they run densely inside, and then sparsely. That is, for a long solenoid on the outside we will assume =0, and inside the solenoid =const. Inside the long solenoid, well, in the vicinity. Let's say, in its middle, the magnetic field is almost uniform, and outside the solenoid this field is small. Then we can find this magnetic field inside as follows: here I take such a contour ( Fig.7.13), and now we write:
1)

.

- this is a full charge. This surface is pierced by turns

(full charge)=
(number of turns piercing this surface).

We get this equality from our law:
, or

Field at a large distance from a limited current distribution.

Magnetic moment

This means that currents flow in a limited region of space, then there is a simple recipe for finding the magnetic field that creates this limited distribution. Well, by the way, any source falls under this concept of limited space, so there is no narrowing here.

If the characteristic size of the system , That
. Let me remind you that we solved a similar problem for the electric field created by a limited charge distribution, and there the concept of a dipole moment and moments of higher order appeared. I will not solve this problem here.

P By analogy (as was done in electrostatics), it can be shown that the magnetic field from a limited distribution over large distances is similar to the electric field of a dipole. That is, the structure of this field is as follows:

The distribution is characterized by a magnetic moment .Magnetic moment
, Where – current density or, if we take into account that we are dealing with moving charged particles, then we can express this formula for a continuous medium in terms of particle charges in this way:
. What does this amount represent? I repeat, the current distribution is created by the movement of these charged particles. Radius vector i-th particle is vectorially multiplied by the speed i-th particle and all this is multiplied by the charge of this i-th particles.

By the way, we had such a design in mechanics. If instead of a charge without a multiplier write the mass of the particle, what will it represent? Momentum of the system.

If we have particles of the same type (
, for example, electrons), then we can write

. This means that if the current is created by particles of the same type, then the magnetic moment is simply related to the angular momentum of this system of particles.

A magnetic field, created by this magnetic moment is equal to:

(8.1 )

Magnetic moment of a turn with current

P Let us say that we have a coil and a current of force flows through it. Vector different from zero within the turn. Let's take an element of this turn ,
, Where S is the cross section of the turn, and – unit tangent vector. Then the magnetic moment is defined as follows:
. What is it
? This is a vector directed along the normal vector to the plane of the coil . And the vector product of two vectors is twice the area of the triangle built on these vectors. If dS– area of a triangle built on vectors And , That
. Then we write the magnetic moment equals. Means,

(magnetic moment of a coil with current) = (current strength) (turn area) (normal to turn) 1) .

And now we have the formula ( 8.1 ) is applicable for a coil with current and comparable to what we obtained last time, just to check the formula, since I created this formula by analogy.

Let us have at the origin of coordinates a coil of arbitrary shape through which a current of force  flows, then the field at a point at a distance X equals: (
). For a round turn
,
. In the last lecture, we found the magnetic field of a circular coil with current, at
these formulas are the same.

At large distances from any current distribution, the magnetic field is found according to the formula ( 8.1 ), and this entire distribution is characterized by one vector, which is called the magnetic moment. By the way, the simplest source of a magnetic field is a magnetic moment. For an electric field the simplest source is a monopole, for an electric field the next most complex is an electric dipole, and for a magnetic field it all starts with this dipole or magnetic moment. This, I draw your attention once again, is insofar as these same monopoles do not exist. If there were a monopole, then everything would be the same as in an electric field. And so our simplest source of magnetic field is a magnetic moment, an analogue of an electric dipole. A clear example of a magnetic moment is a permanent magnet. A permanent magnet has a magnetic moment, and at a large distance its field has the following structure:

Force acting on a current-carrying conductor in a magnetic field

We have seen that a charged particle experiences a force equal to
. The current in a conductor is the result of the movement of charged particles of the body, that is, there is no uniformly spread charge in space, the charge is localized in each particle. Current Density
. On i the th particle is acted upon by a force
.

IN select a volume element
and sum up the forces acting on all particles of this volume element
. The force acting on all particles in a given volume element is defined as the current density on the magnetic field and on the size of the volume element. Now let’s rewrite it in differential form:
, from here
- This force density, force acting per unit volume. Then we get the general formula for force:
.

ABOUT Usually current flows through linear conductors; we rarely encounter cases where the current is somehow spread throughout the volume. Although, by the way, the Earth has a magnetic field, but what does this field come from? The source of the field is a magnetic moment, which means that the Earth has a magnetic moment. And this means that that recipe for the magnetic moment shows that there must be some currents inside the Earth, they must necessarily be closed, because there cannot be a stationary open field. Where do these currents come from, what supports them? I am not an expert in terrestrial magnetism. Some time ago there was no specific model of these currents. They could have been induced there at some point and had not yet died out there. In fact, a current can be excited in a conductor, and then it quickly ends due to energy absorption, heat release, and other things. But, when we are dealing with such volumes as the Earth, then the decay time of these currents, once excited by some mechanism, this decay time can be very long and last geological epochs. Maybe that's how it is. Well, let’s say, a small object like the Moon has a very weak magnetic field, which means that it has already died out there, say, the magnetic field of Mars is also much weaker than the field of the Earth, because Mars is smaller than the Earth. What am I talking about? Of course, there are cases when currents flow in volumes, but what we have here on Earth are usually linear conductors, so we will now transform this formula in relation to a linear conductor.

P If there is a linear conductor, the current flows with force. Select a conductor element , the volume of this element dV,
,
. Force acting on a conductor element
perpendicular to the plane of a triangle built on vectors And , that is, directed perpendicular to the conductor, and the total force is found by summation. Here, two formulas solve this problem.

Magnetic moment in an external field

The magnetic moment itself creates a field; now we are not considering its own field, but we are interested in how the magnetic moment behaves when placed in an external magnetic field. The magnetic moment is acted upon by a moment of force equal to
. The moment of force will be directed perpendicular to the board, and this moment will tend to turn the magnetic moment along the line of force. Why does the compass needle point to the north pole? She, of course, does not care about the geographic pole of the Earth; the compass needle is oriented along the magnetic field line, which, due to random reasons, by the way, is directed approximately along the meridian. Due to what? And the moment acts on her. When an arrow, a magnetic moment coinciding in direction with the arrow itself, does not coincide with the line of force, a moment appears that turns it along this line. Where the magnetic moment of the compass needle comes from, we will discuss this later.

TO In addition, the magnetic moment is acted upon by a force , equal
. If the magnetic moment is directed along , then the force pulls the magnetic moment into the region with higher induction. These formulas are similar to how an electric field acts on a dipole moment; there, too, the dipole moment is oriented along the field and is drawn into a region with higher intensity. Now we can consider the question of the magnetic field in matter.

Magnetic field in matter

A Toms may have magnetic moments. The magnetic moments of atoms are related to the angular momentum of electrons. The formula has already been obtained
, Where – angular momentum of the particle creating the current. In an atom we have a positive nucleus and an electron e, rotating in an orbit, in fact, in due time we will see that this picture has no relation to reality, this is not how we can imagine an electron that rotates, but what remains is that an electron in an atom has an angular momentum, and this angular momentum will correspond such a magnetic moment:
. Visually, a charge rotating in a circle is equivalent to a circular current, that is, it is an elementary coil with current. The angular momentum of an electron in an atom is quantized, that is, it can only take on certain values, according to this recipe:
,
, where is this value is Planck's constant. The angular momentum of an electron in an atom can only take certain values; we will not discuss how this happens now. Well, and as a result of this, the magnetic moment of an atom can take on certain values. These details do not concern us now, but at least we will imagine that an atom can have a certain magnetic moment; there are atoms that do not have a magnetic moment. Then a substance placed in an external field is magnetized, which means that it acquires a certain magnetic moment due to the fact that the magnetic moments of atoms are oriented predominantly along the field.

Volume element dV acquires a magnetic moment
, what does the vector have to do with has the meaning of magnetic moment density and is called the magnetization vector. There is a class of substances called paramagnets, for which
, is magnetized so that the magnetic moment coincides with the direction of the magnetic field. Available diamagnetic materials, which are magnetized, so to speak, “against the grain,” that is, the magnetic moment is antiparallel to the vector , Means,
. This is a more subtle term. What is the vector parallel to the vector It is clear that the magnetic moment of an atom is oriented along the magnetic field. Diamagnetism is related to something else: if an atom does not have a magnetic moment, then in an external magnetic field it acquires a magnetic moment, and the magnetic moment is antiparallel . This very subtle effect is due to the fact that the magnetic field affects the planes of electron orbits, that is, it affects the behavior of angular momentum. The paramagnetic is pulled into the magnetic field, the diamagnetic is pushed out. Now, so that this is not pointless, copper is a diamagnetic, and aluminum is paramagnetic, if you take a magnet, then the aluminum cake will be attracted by the magnet, and then the copper cake will be repelled.

It is clear that the resulting field, when a substance is introduced into a magnetic field, is the sum of the external field and the field created due to the magnetic moment of the substance. Now let's look at the equation
, or in differential form
. Now this statement: magnetization of a substance is equivalent to inducing a current in it with a density
. Then we will write this equation in the form
.

Let's check the dimension: M is the magnetic moment per unit volume
, dimension
. When you write any formula, it is always useful to check the dimension, especially if the formula is your own, that is, you did not copy it, did not remember it, but received it.

N amagnetization is characterized by the vector , it is called the magnetization vector, this is the density of the magnetic moment or the magnetic moment per unit time. I said that magnetization is equivalent to the appearance of current
, the so-called molecular current, and this equation is equivalent to:
, that is, we can assume that there is no magnetization, but there are such currents. Let's set ourselves the following equation:
,- these are real currents associated with specific charge carriers, and these are currents associated with magnetization. An electron in an atom is a circular current, let’s take the area inside, inside the sample all these currents are destroyed, but the presence of such circular currents is equivalent to one total current that flows around this conductor along the surface, hence this formula. Let's rewrite this equation as follows:
,
. This Let's also send it to the left and denote
, vector called magnetic field strength, then the equation takes the form
. (circulation of magnetic field strength along a closed circuit) = (current strength through the surface of this circuit).

Well, and finally, the last thing. We have this formula:
. For many media, magnetization depends on the field strength,
, Where –magnetic susceptibility, is a coefficient characterizing the tendency of a substance to magnetize. Then this formula will be rewritten in the form
,
–magnetic permeability, and we get the following formula:
.

If
, then these are paramagnets,
- these are diamagnetic materials, well, and, finally, there are substances for which this takes large values (about 10 3),
- these are ferromagnets (iron, cobalt and nickel). Ferromagnets are remarkable for this reason. That they are not only magnetized in a magnetic field, but they are characterized by residual magnetization; if it has already been magnetized once, then if the external field is removed, it will remain magnetized, unlike dia- and paramagnets. A permanent magnet is a ferromagnet that is magnetized on its own without an external field. By the way, there are analogues of this matter in electricity: there are dielectrics that are polarized by themselves without any external field. In the presence of matter, our fundamental equation takes on the following form:

A here's another example ferromagnets, a household example of a magnetic field in media, firstly, a permanent magnet, well, and a more subtle thing - a magnetic tape. What is the principle of recording on tape? A tape is a thin tape coated with a ferromagnetic layer, the recording head is a coil with a core through which alternating current flows, an alternating magnetic field is created in the gap, the current tracks the sound signal, oscillations at a certain frequency. Accordingly, in the magnet circuit there is an alternating magnetic field, which changes along with this same current. A ferromagnet is magnetized by alternating current. When this tape is pulled through this type of device, the alternating magnetic field creates an alternating emf. and the electrical signal is played again. These are ferromagnets at the household level.

Let us calculate the field created by a current flowing through a thin straight wire of infinite length.

Magnetic field induction at an arbitrary point A(Fig. 6.12) created by the conductor element d l , will be equal

Rice. 6.12. Magnetic field of a straight conductor

Fields from different elements have the same direction (tangential to a circle with radius R, lying in a plane orthogonal to the conductor). This means we can add (integrate) absolute values

Let's express r and sin through the integration variable l

Then (6.7) can be rewritten as

Thus,

The picture of the magnetic field lines of an infinitely long straight conductor carrying current is shown in Fig. 6.13.

Rice. 6.13. Magnetic field lines of a straight conductor carrying current:
1 - side view; 2, 3 - section of the conductor by a plane perpendicular to the conductor

Rice. 6.14. Designations for the direction of current in a conductor

To indicate the direction of current in a conductor perpendicular to the plane of the figure, we will use the following notation (Fig. 6.14):

Let us recall the expression for the electric field strength of a thin thread charged with a linear charge density

The similarity of expressions is obvious: we have the same dependence on the distance to the thread (current), the linear charge density has been replaced by current strength. But the directions of the fields are different. For a thread, the electric field is directed along the radii. The magnetic field lines of an infinite rectilinear conductor carrying current form a system of concentric circles surrounding the conductor. The directions of the power lines form a right-handed system with the direction of the current.

In Fig. Figure 6.15 presents an experiment in studying the distribution of magnetic field lines around a straight conductor carrying current. A thick copper conductor is passed through holes in a transparent plate on which iron filings are poured. After turning on a direct current of 25 A and tapping on the plate, the sawdust forms chains that repeat the shape of the magnetic field lines.

Around a straight wire perpendicular to the plate, ring lines of force are observed, located most densely near the wire. As you move away from it, the field decreases.

Rice. 6.15. Visualization of magnetic field lines around a straight conductor

In Fig. Figure 6.16 presents experiments to study the distribution of magnetic field lines around wires crossing a cardboard plate. Iron filings poured onto the plate are aligned along the magnetic field lines.

Rice. 6.16. Distribution of magnetic field lines
near the intersection of one, two or several wires with a plate