Magnetic permeability- physical quantity, coefficient (depending on the properties of the medium), characterizing the relationship between magnetic induction texvc not found; See math/README for setup help.): (B) and magnetic field strength Unable to parse expression (executable file texvc not found; See math/README for setup help.): (H) in substance. For different media, this coefficient is different, so they talk about the magnetic permeability of a particular medium (implying its composition, state, temperature, etc.).

First found in the work of Werner Siemens "Beiträge zur Theorie des Elektromagnetismus" ("Contribution to the theory of electromagnetism") in 1881.

Usually denoted by a Greek letter Unable to parse expression (executable file texvc . It can be either a scalar (for isotropic substances) or a tensor (for anisotropic substances).

In general, the relationship between magnetic induction and magnetic field strength through magnetic permeability is introduced as

Unable to parse expression (executable file texvc not found; See math/README for setup help.): \vec(B) = \mu\vec(H),

and Unable to parse expression (executable file texvc not found; See math/README for setup help.): \mu in the general case, here it should be understood as a tensor, which in the component notation corresponds to:

Unable to parse expression (executable file texvc not found; See math/README for setup help.): \ B_i = \mu_(ij)H_j

For isotropic substances, the ratio:

Unable to parse expression (executable file texvc not found; See math/README for setup help.): \vec(B) = \mu\vec(H)

can be understood in the sense of multiplying a vector by a scalar (the magnetic permeability is reduced in this case to a scalar).

Often the designation Unable to parse expression (executable file texvc not found; See math/README for setup help.): \mu is used differently than here, namely for the relative magnetic permeability (in this case Unable to parse expression (executable file texvc not found; See math/README for setup help.): \mu coincides with that in the GHS).

The dimension of absolute magnetic permeability in SI is the same as the dimension of the magnetic constant, that is, H / or / 2 .

The relative magnetic permeability in SI is related to the magnetic susceptibility χ by the relation

Unable to parse expression (executable file texvc not found; See math/README for setup help.): \mu_r = 1 + \chi,

Classification of substances according to the value of magnetic permeability

The vast majority of substances belong either to the class of diamagnets ( Unable to parse expression (executable file texvc not found; See math/README for setup help.): \mu \lessapprox 1), or to the class of paramagnets ( Unable to parse expression (executable file texvc not found; See math/README for setup help.): \mu \gtrapprox 1). But a number of substances - (ferromagnets), for example iron, have more pronounced magnetic properties.

In ferromagnets, due to hysteresis, the concept of magnetic permeability, strictly speaking, is not applicable. However, in a certain range of variation of the magnetizing field (so that the residual magnetization can be neglected, but up to saturation), it is possible, in a better or worse approximation, to represent this dependence as a linear one (and for magnetically soft materials, the limitation from below may not be too significant in practice), and in In this sense, the magnitude of the magnetic permeability can also be measured for them.

Magnetic permeability of some substances and materials

Magnetic susceptibility of some substances

Magnetic susceptibility and magnetic permeability of some materials

Medium	Susceptibility χ m (volumetric, SI)	Permeability μ [H/m]	Relative permeability μ/μ 0	A magnetic field	Maximum frequency
Metglas (English) Metglas )		1,25	1 000 000	at 0.5 T	100 kHz
Nanoperm (English) Nanoperm )		10×10 -2	80 000	at 0.5 T	10 kHz
mu metal		2.5×10 -2	20 000	at 0.002 T
mu metal			50 000
Permalloy		1.0×10 -2	70 000	at 0.002 T
electrical steel		5.0×10 -3	4000	at 0.002 T
Ferrite (nickel-zinc)		2.0×10 -5 - 8.0×10 -4	16-640		100 kHz ~ 1 MHz [[C:Wikipedia:Articles without sources (country: Lua error: callParserFunction: function "#property" was not found. )]][[C:Wikipedia:Articles without sources (country: Lua error: callParserFunction: function "#property" was not found. )]]
Ferrite (manganese-zinc)		>8.0×10 -4	640 (and more)		100 kHz ~ 1 MHz
Steel		8.75×10 -4	100	at 0.002 T
Nickel		1.25×10 -4	100 - 600	at 0.002 T
Neodymium magnet			1.05	up to 1.2-1.4 T
Platinum		1.2569701×10 -6	1,000265
Aluminum	2.22×10 -5	1.2566650×10 -6	1,000022
Wood			1,00000043
Air			1,00000037
Concrete			1
Vacuum	0	1.2566371×10 -6 (μ 0)	1
Hydrogen	-2.2×10 -9	1.2566371×10 -6	1,0000000
Teflon		1.2567×10 -6	1,0000
Sapphire	-2.1×10 -7	1.2566368×10 -6	0,99999976
Copper	-6.4×10 -6 or -9.2×10 -6	1.2566290×10 -6	0,999994
Water	-8.0×10 -6	1.2566270×10 -6	0,999992
Bismuth	-1.66×10 -4		0,999834
superconductors	−1	0	0

see also

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Notes

An excerpt characterizing Magnetic permeability

I felt so sorry for him!.. But, unfortunately, it was not in my power to help him. And honestly, I really wanted to know how this extraordinary baby helped him ...
– We found them! Stella repeated again. – I didn’t know how to do it, but my grandmother helped me!
It turned out that Harold, during his lifetime, did not even have time to find out how terribly his family suffered when dying. He was a warrior knight, and died before his city was in the hands of the "executioners", as his wife predicted.
But, as soon as he got into this, unfamiliar, marvelous world of "departed" people, he could immediately see how ruthlessly and cruelly evil fate dealt with his "only and beloved". After that, like a man possessed, for an eternity he tried somehow, somewhere, to find these people, the most dear to him in the whole wide world ... And he searched for them for a very long time, more than a thousand years, until one day some, completely unfamiliar, sweet girl Stella did not offer him to "make him happy" and did not open that "other" right door to finally find them for him...
- Do you want me to show you? - again suggested the baby,
But I was no longer so sure if I wanted to see something else ... Because the visions she had just shown hurt my soul, and it was impossible to get rid of them so quickly to want to see some kind of continuation ...
“But you want to see what happened to them!” - confidently stated the "fact" little Stella.
I looked at Harold and saw in his eyes the complete understanding of what I had just experienced unexpectedly.
– I know what you saw... I watched it many times. But now they are happy, we go to look at them very often... And the "former" ones too... - the "sad knight" said quietly.
And then only I realized that Stella, simply, when he wanted it, transferred him to his own past, just like she had just done it !!! And she did it almost effortlessly! .. I didn’t even notice how this wonderful, bright girl began to “attach” me to herself more and more, becoming for me almost a real miracle, which I endlessly wanted to watch ... And which I didn’t want to leave at all ... Then I knew almost nothing and didn’t know how, except for what I could understand and learn myself, and I really wanted to learn at least something from her, while there was still such an opportunity.
- Come to me, please! - Stella, suddenly saddened, whispered softly, - you know that you still can’t stay here ... Grandmother said that you won’t stay for a very, very long time ... That you still can’t die. But you come...
Everything around suddenly became dark and cold, as if black clouds suddenly covered such a colorful and bright Stella's world...
“Oh, don’t think about such a terrible thing! - the girl was indignant, and, like an artist with a brush on the canvas, she quickly “painted over” everything again in a bright and joyful color.
- Well, is it really better? she asked rather.
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- Yes, of course! Stella laughed. - You are strong, so you create everything around you in your own way.
– But how then to think? .. – I still could not “drive” into the incomprehensible self.
“And you just “close” and show only what you want to show,” my amazing friend said as a matter of course. “Grandma taught me that.
I thought that, apparently, the time had come for me to “shake” my “secret” grandmother a little, who (I was almost sure of it!) Probably knew something, but for some reason did not want to teach me anything yet .. .
"So you want to see what happened to Harold's family?" the little girl asked impatiently.
To be honest, I didn’t have too much desire, since I wasn’t sure what to expect from this “show”. But in order not to offend the generous Stella, she agreed.
I won't show you for a long time. Promise! But you should know about them, right? .. - the girl said in a happy voice. - Look, the son will be the first ...

To my great surprise, unlike what I had seen before, we ended up in a completely different time and place, which was similar to France, and in clothes resembled the eighteenth century. A covered beautiful carriage drove along a wide cobbled street, inside of which sat a young man and a woman in very expensive suits, and apparently in a very bad mood ... The young man stubbornly proved something to the girl, and she, completely not listening to him, calmly hovered somewhere in my daydreams, which irritated the young man very much ...
- You see, it's him! This is the same "little boy" ... only after many, many years, - Stella whispered softly.
"How do you know it's really him?" – Still not quite understanding, I asked.
- Well, it's very simple! The little girl looked at me in surprise. - We all have an essence, and the essence has its own “key”, by which each of us can be found, you just need to know how to look. Here look...
She showed me the baby again, Harold's son.
“Think of its essence, and you will see...
And I immediately saw a transparent, brightly glowing, surprisingly powerful entity, on the chest of which an unusual “diamond” energy star was burning. This "star" shone and shimmered with all the colors of the rainbow, now decreasing, then increasing, as if slowly pulsing, and sparkled so brightly, as if it really had been created from the most amazing diamonds.
“You see that strange upside-down star on his chest?” This is his key. And if you try to follow him like a thread, then it will lead you straight to Axel, who has the same star - this is the same essence, only in its next incarnation.
I looked at her with all my eyes, and apparently noticing this, Stella laughed and cheerfully admitted:
- Don't think that it's me myself - it was my grandmother who taught me! ..
I was very ashamed to feel like a complete bum, but the desire to know more was a hundred times stronger than any shame, so I hid my pride as deep as possible and carefully asked:
– And what about all these amazing “realities” that we are now seeing here? After all, this is someone else's, specific life, and you do not create them in the same way as you create all your worlds?
- Oh no! - again, the baby was delighted with the opportunity to explain something to me. - Of course not! It's just the past in which all these people once lived, and I'm just taking you and me there.
- And Harold? How does he see all this?
Oh, it's easy for him! He's just like me, dead, so he can move wherever he wants. After all, he no longer has a physical body, so his essence knows no obstacles here and can walk wherever she wants ... just like me ... - the little girl finished sadly.
I sadly thought that what was for her just a “simple transfer to the past”, for me, apparently, will be a “mystery behind seven locks” for a long time ... But Stella, as if having heard my thoughts, immediately hurried to reassure me :
- You'll see, it's very simple! You just have to try.
- And these "keys", don't they ever repeat with others? I decided to continue my questions.
- No, but sometimes something else happens ... - for some reason, smiling funny, the baby answered. - At the beginning, that’s exactly how I got caught, for which I was very much “beaten” ... Oh, it was so stupid! ..
- But as? I asked very interested.
Stella replied cheerfully:
- Oh, that was very funny! - and after a little thought, she added, - but it’s also dangerous ... I searched all the “floors” for the past incarnation of my grandmother, and instead of her, along her “thread”, a completely different entity came, which somehow managed to “copy” my grandmother’s “ flower” (apparently also a “key”!) and, as soon as I managed to be glad that I had finally found it, this unfamiliar entity mercilessly hit me in the chest. Yes, so much so that my soul almost flew away! ..
"But how did you get rid of her?" I was surprised.
- Well, to be honest, I didn’t get rid of it ... - the girl was embarrassed. - I just called my grandmother ...
What do you call "floors"? I still couldn't calm down.
– Well, these are different “worlds” where the spirits of the dead live... In the most beautiful and highest, those who were good live... and, probably, the strongest too.
- People like you? I asked smiling.
– Oh, no, of course! I must have gotten here by mistake. - The girl said sincerely. – Do you know what is the most interesting? From this "floor" we can walk everywhere, but from the others no one can get here ... Is it really interesting? ..
Yes, it was very strange and very exciting for my “hungry” brain, and I really wanted to know more! gave me something (like, for example, my “star friends”), and therefore, even such a simple childish explanation already made me extraordinarily happy and made me delve even more furiously into my experiments, conclusions and mistakes ... as usual, finding in everything that happens even more incomprehensible. My problem was that I could do or create “unusual” very easily, but the whole trouble was that I also wanted to understand how I create it all ... Namely, this is what I have not been very successful with yet ...

Magnetics

All substances in a magnetic field are magnetized (an internal magnetic field arises in them). Depending on the magnitude and direction of the internal field, substances are divided into:

1) diamagnets,

2) paramagnets,

3) ferromagnets.

The magnetization of a substance is characterized by magnetic permeability,

Magnetic induction in matter,

Magnetic induction in vacuum.

Any atom can be characterized by a magnetic moment .

The current in the circuit, - the area of ​​the circuit, - the vector of the normal to the surface of the circuit.

The microcurrent of an atom is created by the movement of negative electrons along the orbit and around its own axis, as well as by the rotation of the positive nucleus around its own axis.

1. Diamagnets.

When there is no external field, in atoms diamagnets electron and nucleus currents are compensated. The total microcurrent of an atom and its magnetic moment are equal to zero.

In an external magnetic field, nonzero elementary currents are induced (induced) in atoms. In this case, the magnetic moments of the atoms are oriented oppositely.

A small own field is created, directed oppositely to the external one, and weakening it.

in diamagnets.

Because< , то для диамагнетиков 1.

2. Paramagnets

AT paramagnets microcurrents of atoms and their magnetic moments are not equal to zero.

Without an external field, these microcurrents are located randomly.

In an external magnetic field, the microcurrents of paramagnetic atoms are oriented along the field, amplifying it.

In a paramagnet, the magnetic induction = + slightly exceeds .

For paramagnets, 1. For dia- and paramagnets, you can count 1.

Table 1. Magnetic permeability of para- and diamagnets.

The magnetization of paramagnets depends on temperature, because. the thermal motion of atoms prevents the ordered arrangement of microcurrents.

Most substances in nature are paramagnetic.

The intrinsic magnetic field in dia- and paramagnets is insignificant and is destroyed if the substance is removed from the external field (the atoms return to their original state, the substance is demagnetized).

3. Ferromagnets

Magnetic permeability ferromagnets reaches hundreds of thousands and depends on the magnitude of the magnetizing field ( highly magnetic substances).

Ferromagnets: iron, steel, nickel, cobalt, their alloys and compounds.

In ferromagnets, there are regions of spontaneous magnetization ("domains") in which all microcurrents of atoms are oriented in the same way. The domain size reaches 0.1 mm.

In the absence of an external field, the magnetic moments of individual domains are randomly oriented and compensate. In the external field, those domains in which microcurrents enhance the external field increase their size at the expense of neighboring ones. The resulting magnetic field = + in ferromagnets is much stronger than in para- and diamagnets.

Domains containing billions of atoms have inertia and do not quickly return to their original disordered state. Therefore, if a ferromagnet is removed from the external field, then its own field is preserved for a long time.

The magnet demagnetizes during long-term storage (over time, the domains return to a chaotic state).

Another method of demagnetization is heating. For each ferromagnet, there is a temperature (it is called the “Curie point”) at which bonds between atoms are destroyed in the domains. In this case, the ferromagnet turns into a paramagnet and demagnetization occurs. For example, the Curie point for iron is 770°C.

called magnetic permeability . Absolute magneticpermeability environment is the ratio of B to H. According to the International System of Units, it is measured in units called 1 henry per meter.

Its numerical value is expressed by the ratio of its value to the value of the magnetic permeability of the vacuum and is denoted by µ. This value is called relative magneticpermeability(or simply magnetic permeability) of the medium. As a relative quantity, it has no unit of measure.

Therefore, the relative magnetic permeability µ is a value showing how many times the field induction of a given medium is less (or more) than the vacuum magnetic field induction.

When a substance is exposed to an external magnetic field, it becomes magnetized. How does this happen? According to Ampere's hypothesis, microscopic electric currents constantly circulate in every substance, caused by the movement of electrons in their orbits and the presence of their own. Under normal conditions, this movement is disordered, and the fields “quench” (compensate) each other. When a body is placed in an external field, the currents are ordered, and the body becomes magnetized (that is, it has its own field).

The magnetic permeability of all substances is different. Based on its size, substances are subject to division into three large groups.

At diamagnets the value of the magnetic permeability µ is slightly less than unity. For example, bismuth has µ = 0.9998. Diamagnets include zinc, lead, quartz, copper, glass, hydrogen, benzene, and water.

Magnetic permeability paramagnets slightly more than unity (for aluminum, µ = 1.000023). Examples of paramagnets are nickel, oxygen, tungsten, ebonite, platinum, nitrogen, air.

Finally, the third group includes a number of substances (mainly metals and alloys), whose magnetic permeability significantly (by several orders of magnitude) exceeds unity. These substances are ferromagnets. These mainly include nickel, iron, cobalt and their alloys. For steel µ = 8∙10^3, for nickel-iron alloy µ=2.5∙10^5. Ferromagnets have properties that distinguish them from other substances. First, they have residual magnetism. Secondly, their magnetic permeability depends on the magnitude of the induction of the external field. Thirdly, for each of them there is a certain temperature threshold, called Curie point, at which it loses its ferromagnetic properties and becomes a paramagnet. For nickel the Curie point is 360°C, for iron it is 770°C.

The properties of ferromagnets are determined not only by the magnetic permeability, but also by the value of I, called magnetization of this substance. This is a complex nonlinear function of magnetic induction, the growth of magnetization is described by a line called magnetization curve. In this case, having reached a certain point, the magnetization practically stops growing (there comes magnetic saturation). The lag of the value of the magnetization of a ferromagnet from the growing value of the induction of the external field is called magnetic hysteresis. In this case, there is a dependence of the magnetic characteristics of a ferromagnet not only on its current state, but also on its previous magnetization. The graphic representation of the curve of this dependence is called hysteresis loop.

Due to their properties, ferromagnets are widely used in engineering. They are used in the rotors of generators and electric motors, in the manufacture of transformer cores and in the production of parts for electronic computers. ferromagnets are used in tape recorders, telephones, magnetic tapes and other media.

The magnetic field of the coil is determined by the current and the intensity of this field, and the field induction. Those. the field induction in vacuum is proportional to the magnitude of the current. If a magnetic field is created in a certain medium or substance, then the field acts on the substance, and it, in turn, changes the magnetic field in a certain way.

A substance in an external magnetic field becomes magnetized and an additional internal magnetic field arises in it. It is associated with the movement of electrons along intraatomic orbits, as well as around their own axis. The motion of electrons and nuclei of atoms can be considered as elementary circular currents.

The magnetic properties of an elementary circular current are characterized by a magnetic moment.

In the absence of an external magnetic field, the elementary currents inside the substance are oriented randomly (chaotically) and, therefore, the total or total magnetic moment is zero and the magnetic field of elementary internal currents is not detected in the surrounding space.

The effect of an external magnetic field on elementary currents in matter is that the orientation of the axes of rotation of charged particles changes so that their magnetic moments turn out to be directed in one direction. (toward the external magnetic field). The intensity and nature of magnetization in different substances in the same external magnetic field differ significantly. The value characterizing the properties of the medium and the influence of the medium on the magnetic field density is called absolute magnetic permeability or magnetic permeability of the medium (μ With ) . This is the relation = . Measured [ μ With ]=H/m.

The absolute magnetic permeability of vacuum is called the magnetic constant μ about \u003d 4π 10 -7 Gn / m.

The ratio of the absolute magnetic permeability to the magnetic constant is called relative magnetic permeabilityμ c /μ 0 \u003d μ. Those. relative magnetic permeability is a value showing how many times the absolute magnetic permeability of a medium is greater or less than the absolute permeability of vacuum. μ is a dimensionless quantity that varies over a wide range. This value is the basis for dividing all materials and media into three groups.

Diamagnets . These substances have μ< 1. К ним относятся - медь, серебро, цинк, ртуть, свинец, сера, хлор, вода и др. Например, у меди μ Cu = 0,999995. Эти вещества слабо взаимодействуют с магнитом.

Paramagnets . These substances have μ > 1. These include aluminum, magnesium, tin, platinum, manganese, oxygen, air, etc. Air has = 1.0000031. . These substances, as well as diamagnets, weakly interact with a magnet.

For technical calculations, μ of diamagnetic and paramagnetic bodies is assumed to be equal to one.

ferromagnets . This is a special group of substances that play a huge role in electrical engineering. These substances have μ >> 1. These include iron, steel, cast iron, nickel, cobalt, gadolinium and metal alloys. These substances are strongly attracted to a magnet. These substances have μ = 600-10,000. For some alloys, μ reaches record values ​​up to 100,000. It should be noted that μ for ferromagnetic materials is not constant and depends on the magnetic field strength, type of material and temperature.

The large value of µ in ferromagnets is explained by the fact that they have regions of spontaneous magnetization (domains), within which the elementary magnetic moments are directed in the same way. When added together, they form the common magnetic moments of the domains.

In the absence of a magnetic field, the magnetic moments of the domains are randomly oriented and the total magnetic moment of the body or substance is zero. Under the action of an external field, the magnetic moments of the domains are oriented in one direction and form the total magnetic moment of the body, directed in the same direction as the external magnetic field.

This important feature is used in practice, using ferromagnetic cores in coils, which makes it possible to sharply increase the magnetic induction and magnetic flux at the same values ​​of currents and the number of turns, or, in other words, to concentrate the magnetic field in a relatively small volume.

Determination of the magnetic permeability of a substance. Its role in the description of the magnetic field

If you conduct an experiment with a solenoid that is connected to a ballistic galvanometer, then when the current is turned on in the solenoid, you can determine the value of the magnetic flux Ф, which will be proportional to the rejection of the galvanometer needle. We will conduct the experiment twice, and we will set the current (I) in the galvanometer to be the same, but in the first experiment the solenoid will be without a core, and in the second experiment, before turning on the current, we will introduce an iron core into the solenoid. It is found that in the second experiment the magnetic flux is significantly greater than in the first (without a core). When repeating the experiment with cores of different thicknesses, it turns out that the maximum flux is obtained when the entire solenoid is filled with iron, that is, the winding is tightly wound around the iron core. You can experiment with different cores. The result is that:

where $Ф$ is the magnetic flux in a coil with a core, $Ф_0$ is the magnetic flux in a coil without a core. The increase in the magnetic flux when the core is introduced into the solenoid is explained by the fact that the magnetic flux created by a combination of oriented ampere molecular currents was added to the magnetic flux, which creates a current in the solenoid winding. Under the influence of a magnetic field, molecular currents are oriented, and their total magnetic moment ceases to be equal to zero, an additional magnetic field arises.

Definition

The value $\mu $, which characterizes the magnetic properties of the medium, is called the magnetic permeability (or relative magnetic permeability).

This is a dimensionless characteristic of matter. An increase in the flux Ф by $\mu $ times (1) means that the magnetic induction $\overrightarrow(B)$ in the core is as many times greater than in vacuum at the same current in the solenoid. Therefore, it can be written that:

\[\overrightarrow(B)=\mu (\overrightarrow(B))_0\left(2\right),\]

where $(\overrightarrow(B))_0$ is the magnetic field induction in vacuum.

Along with magnetic induction, which is the main force characteristic of the field, such an auxiliary vector quantity as the magnetic field strength ($\overrightarrow(H)$) is used, which is related to $\overrightarrow(B)$ by the following relationship:

\[\overrightarrow(B)=\mu \overrightarrow(H)\left(3\right).\]

If formula (3) is applied to the experiment with a core, then we get that in the absence of a core:

\[(\overrightarrow(B))_0=(\mu )_0\overrightarrow(H_0)\left(4\right),\]

where $\mu$=1. In the presence of a core, we get:

\[\overrightarrow(B)=\mu (\mu )_0\overrightarrow(H)\left(5\right).\]

But since (2) is satisfied, it turns out that:

\[\mu (\mu )_0\overrightarrow(H)=(\mu m)_0\overrightarrow(H_0)\to \overrightarrow(H)=\overrightarrow(H_0)\left(6\right).\]

We have obtained that the strength of the magnetic field does not depend on what kind of homogeneous substance the space is filled with. The magnetic permeability of most substances is about unity, with the exception of ferromagnets.

Magnetic susceptibility of matter

Usually, the magnetization vector ($\overrightarrow(J)$) is associated with the intensity vector at each point of the magnet:

\[\overrightarrow(J)=\varkappa \overrightarrow(H)\left(7\right),\]

where $\varkappa $ is the magnetic susceptibility, a dimensionless quantity. For non-ferromagnetic substances and in small fields, $\varkappa $ does not depend on the intensity, it is a scalar quantity. In anisotropic media $\varkappa$ is a tensor and the directions of $\overrightarrow(J)$ and $\overrightarrow(H)$ do not coincide.

Relationship between magnetic susceptibility and magnetic permeability

\[\overrightarrow(H)=\frac(\overrightarrow(B))((\mu )_0)-\overrightarrow(J)\left(8\right).\]

Substitute in (8) the expression for the magnetization vector (7), we get:

\[\overrightarrow(H)=\frac(\overrightarrow(B))((\mu )_0)-\overrightarrow(H)\left(9\right).\]

We express the tension, we get:

\[\overrightarrow(H)=\frac(\overrightarrow(B))((\mu )_0\left(1+\varkappa \right))\to \overrightarrow(B)=(\mu )_0\left( 1+\varkappa \right)\overrightarrow(H)\left(10\right).\]

Comparing expressions (5) and (10), we get:

\[\mu =1+\varkappa \left(11\right).\]

Magnetic susceptibility can be either positive or negative. From (11) it follows that the magnetic permeability can be both greater than unity and less than it.

Example 1

Task: Calculate the magnetization at the center of a circular coil of radius R=0.1 m with a current of I=2A if it is immersed in liquid oxygen. The magnetic susceptibility of liquid oxygen is $\varkappa =3.4\cdot (10)^(-3).$

As a basis for solving the problem, we take an expression that reflects the relationship between the magnetic field strength and magnetization:

\[\overrightarrow(J)=\varkappa \overrightarrow(H)\left(1.1\right).\]

Let's find the field in the center of the coil with current, since we need to calculate the magnetization at this point.

We select an elementary section on the conductor with current (Fig. 1), as the basis for solving the problem, we use the formula for the intensity of the element of the coil with current:

where $\ \overrightarrow(r)$ is the radius vector drawn from the current element to the point under consideration, $\overrightarrow(dl)$ is the conductor element with current (the direction is given by the current direction), $\vartheta$ is the angle between $ \overrightarrow(dl)$ and $\overrightarrow(r)$. Based on Fig. 1 $\vartheta=90()^\circ $, therefore (1.1) will be simplified, in addition, the distance from the center of the circle (the point where we are looking for the magnetic field) of the conductor element with current is constant and equal to the radius of the coil (R), therefore we have:

The resulting vector of the magnetic field strength is directed along the X axis, it can be found as the sum of individual vectors $\ \ \overrightarrow(dH),$ since all current elements create magnetic fields in the center of the wick, directed along the normal of the coil. Then, according to the principle of superposition, the total strength of the magnetic field can be obtained by going to the integral:

We substitute (1.3) into (1.4), we get:

We find the magnetization, if we substitute the intensity from (1.5) into (1.1), we get:

All units are given in the SI system, let's do the calculations:

Answer: $J=3,4\cdot (10)^(-2)\frac(A)(m).$

Example 2

Task: Calculate the proportion of the total magnetic field in a tungsten rod, which is in an external uniform magnetic field, which is determined by molecular currents. The magnetic permeability of tungsten is $\mu =1.0176.$

The magnetic field induction ($B"$), which is accounted for by molecular currents, can be found as:

where $J$ is the magnetization. It is related to the magnetic field strength by the expression:

where the magnetic susceptibility of a substance can be found as:

\[\varkappa =\mu -1\ \left(2.3\right).\]

Therefore, we find the magnetic field of molecular currents as:

The total field in the bar is calculated according to the formula:

We use expressions (2.4) and (2.5) to find the required relation:

\[\frac(B")(B)=\frac((\mu )_0\left(\mu -1\right)H)(\mu (\mu )_0H)=\frac(\mu -1) (\mu ).\]

Let's do the calculations:

\[\frac(B")(B)=\frac(1.0176-1)(1.0176)=0.0173.\]

Answer: $\frac(B")(B)=0.0173.$