A magnetic field is a special form of matter that is created by magnets, conductors with current (moving charged particles) and which can be detected by the interaction of magnets, conductors with current (moving charged particles).
Oersted's experience
The first experiments (carried out in 1820) that showed that there is a deep connection between electrical and magnetic phenomena were the experiments of the Danish physicist H. Oersted.
A magnetic needle located near a conductor rotates through a certain angle when the current in the conductor is turned on. When the circuit is opened, the arrow returns to its original position.
From the experience of G. Oersted it follows that there is a magnetic field around this conductor.
Ampere's experience
Two parallel conductors through which electric current flows interact with each other: they attract if the currents are in the same direction, and repel if the currents are in the opposite direction. This occurs due to the interaction of magnetic fields arising around the conductors.
Properties of magnetic field
1. Materially, i.e. exists independently of us and our knowledge about it.
2. Created by magnets, conductors with current (moving charged particles)
3. Detected by the interaction of magnets, conductors with current (moving charged particles)
4. Acts on magnets, current-carrying conductors (moving charged particles) with some force
5. There are no magnetic charges in nature. You cannot separate the north and south poles and get a body with one pole.
6. The reason why bodies have magnetic properties was found by the French scientist Ampere. Ampere put forward the conclusion that the magnetic properties of any body are determined by closed electric currents inside it.
These currents represent the movement of electrons around orbits in an atom.
If the planes in which these currents circulate are located randomly in relation to each other due to the thermal movement of the molecules that make up the body, then their interactions are mutually compensated and the body does not exhibit any magnetic properties.
And vice versa: if the planes in which the electrons rotate are parallel to each other and the directions of the normals to these planes coincide, then such substances enhance the external magnetic field.
7. Magnetic forces act in a magnetic field in certain directions, which are called magnetic lines of force. With their help, you can conveniently and clearly show the magnetic field in a particular case.
In order to more accurately depict the magnetic field, it was agreed that in those places where the field is stronger, the field lines should be shown denser, i.e. closer to each other. And vice versa, in places where the field is weaker, fewer field lines are shown, i.e. less frequently located.
8. The magnetic field is characterized by the magnetic induction vector.
The magnetic induction vector is a vector quantity characterizing the magnetic field.
The direction of the magnetic induction vector coincides with the direction of the north pole of the free magnetic needle at a given point.
The direction of the field induction vector and current strength I are related by the “right screw (gimlet) rule”:
if you screw in a gimlet in the direction of the current in the conductor, then the direction of the speed of movement of the end of its handle at a given point will coincide with the direction of the magnetic induction vector at that point.
Magnetic field- this is the material medium through which interaction occurs between conductors with current or moving charges.
Properties of magnetic field:
Characteristics of the magnetic field:
To study the magnetic field, a test circuit with current is used. It is small in size, and the current in it is much less than the current in the conductor creating the magnetic field. On opposite sides of the current-carrying circuit, forces from the magnetic field act that are equal in magnitude, but directed in opposite directions, since the direction of the force depends on the direction of the current. The points of application of these forces do not lie on the same straight line. Such forces are called a couple of forces. As a result of the action of a pair of forces, the circuit cannot move translationally; it rotates around its axis. The rotating action is characterized torque.
, Where l–leverage couple of forces(distance between points of application of forces).
As the current in the test circuit or the area of the circuit increases, the torque of the pair of forces will increase proportionally. The ratio of the maximum moment of force acting on the circuit with current to the magnitude of the current in the circuit and the area of the circuit is a constant value for a given point in the field. It's called magnetic induction.
, Where 
-magnetic moment circuit with current.
Unit of measurement magnetic induction – Tesla [T].
Magnetic moment of the circuit– vector quantity, the direction of which depends on the direction of the current in the circuit and is determined by right screw rule: clench your right hand into a fist, point four fingers in the direction of the current in the circuit, then the thumb will indicate the direction of the magnetic moment vector. The magnetic moment vector is always perpendicular to the contour plane.
For direction of the magnetic induction vector take the direction of the vector of the magnetic moment of the circuit, oriented in the magnetic field.
Magnetic induction line– a line whose tangent at each point coincides with the direction of the magnetic induction vector. Magnetic induction lines are always closed and never intersect. Magnetic induction lines of a straight conductor with current have the form of circles located in a plane perpendicular to the conductor. The direction of the magnetic induction lines is determined by the right-hand screw rule. Magnetic induction lines of circular current(turns with current) also have the form of circles. Each coil element is length 
can be imagined as a straight conductor that creates its own magnetic field. For magnetic fields, the principle of superposition (independent addition) applies. The total vector of magnetic induction of the circular current is determined as the result of the addition of these fields in the center of the turn according to the right-hand screw rule.
If the magnitude and direction of the magnetic induction vector are the same at every point in space, then the magnetic field is called homogeneous. If the magnitude and direction of the magnetic induction vector at each point do not change over time, then such a field is called permanent.
Magnitude magnetic induction at any point in the field is directly proportional to the current strength in the conductor creating the field, inversely proportional to the distance from the conductor to a given point in the field, depends on the properties of the medium and the shape of the conductor creating the field.
, Where 
N/A 2 ; Gn/m – magnetic constant of vacuum,
-relative magnetic permeability of the medium,
-absolute magnetic permeability of the medium.
Depending on the value of magnetic permeability, all substances are divided into three classes:
As the absolute permeability of the medium increases, the magnetic induction at a given point in the field also increases. The ratio of magnetic induction to the absolute magnetic permeability of the medium is a constant value for a given poly point, e is called tension.
.
The vectors of tension and magnetic induction coincide in direction. The magnetic field strength does not depend on the properties of the medium.
Ampere power– the force with which the magnetic field acts on a current-carrying conductor.
Where l– length of the conductor, 
- the angle between the magnetic induction vector and the direction of the current.
The direction of the Ampere force is determined by left hand rule: the left hand is positioned so that the component of the magnetic induction vector, perpendicular to the conductor, enters the palm, four extended fingers are directed along the current, then the thumb bent by 90 0 will indicate the direction of the Ampere force.
The result of the Ampere force is the movement of the conductor in a given direction.
E 
if 
= 90 0 , then F=max, if 
= 0 0 , then F = 0.
Lorentz force– the force of the magnetic field on a moving charge.
, where q is the charge, v is the speed of its movement, 
- the angle between the vectors of tension and speed.
The Lorentz force is always perpendicular to the magnetic induction and velocity vectors. The direction is determined by left hand rule(fingers follow the movement of the positive charge). If the direction of the particle's velocity is perpendicular to the magnetic induction lines of a uniform magnetic field, then the particle moves in a circle without changing its kinetic energy.
Since the direction of the Lorentz force depends on the sign of the charge, it is used to separate charges.
Magnetic flux– a value equal to the number of magnetic induction lines that pass through any area located perpendicular to the magnetic induction lines.
, Where 
- the angle between the magnetic induction and the normal (perpendicular) to the area S.
Unit of measurement– Weber [Wb].
Magnetic flux measurement methods:
Changing the orientation of the site in a magnetic field (changing the angle)
Changing the area of a circuit placed in a magnetic field
Change in current strength creating a magnetic field
Changing the distance of the circuit from the magnetic field source
Changes in the magnetic properties of the medium.
F 
Araday recorded an electric current in a circuit that did not contain a source, but was located next to another circuit containing a source. Moreover, the current in the first circuit arose in the following cases: with any change in the current in circuit A, with relative movement of the circuits, with the introduction of an iron rod into circuit A, with the movement of a permanent magnet relative to circuit B. Directed movement of free charges (current) occurs only in an electric field. This means that a changing magnetic field generates an electric field, which sets in motion the free charges of the conductor. This electric field is called induced or vortex.
Differences between a vortex electric field and an electrostatic one:
The source of the vortex field is a changing magnetic field.
The vortex field intensity lines are closed.
The work done by this field to move a charge along a closed circuit is not zero.
The energy characteristic of a vortex field is not the potential, but induced emf– a value equal to the work of external forces (forces of non-electrostatic origin) to move a unit of charge along a closed circuit.
.Measured in Volts[IN].
A vortex electric field occurs with any change in the magnetic field, regardless of whether there is a conducting closed circuit or not. The circuit only allows one to detect the vortex electric field.
Electromagnetic induction- this is the occurrence of induced emf in a closed circuit with any change in the magnetic flux through its surface.
The induced emf in a closed circuit generates an induced current.
.
Direction of induction current determined by Lenz's rule: the induced current is in such a direction that the magnetic field created by it counteracts any change in the magnetic flux that generated this current.
Faraday's law for electromagnetic induction: The induced emf in a closed loop is directly proportional to the rate of change of magnetic flux through the surface bounded by the loop.
T 
oki fuko– eddy induction currents that arise in large conductors placed in a changing magnetic field. The resistance of such a conductor is low, since it has a large cross-section S, so the Foucault currents can be large in value, as a result of which the conductor heats up.
Self-induction- this is the occurrence of induced emf in a conductor when the current strength in it changes.
A conductor carrying current creates a magnetic field. Magnetic induction depends on the current strength, therefore the intrinsic magnetic flux also depends on the current strength.
, where L is the proportionality coefficient, inductance.
Unit of measurement inductance – Henry [H].
Inductance conductor depends on its size, shape and magnetic permeability of the medium.
Inductance increases with increasing length of the conductor, the inductance of a turn is greater than the inductance of a straight conductor of the same length, the inductance of a coil (a conductor with a large number of turns) is greater than the inductance of one turn, the inductance of a coil increases if an iron rod is inserted into it.
Faraday's law for self-induction:
.
Self-induced emf is directly proportional to the rate of change of current.
Self-induced emf generates a self-induction current, which always prevents any change in the current in the circuit, that is, if the current increases, the self-induction current is directed in the opposite direction; when the current in the circuit decreases, the self-induction current is directed in the same direction. The greater the inductance of the coil, the greater the self-inductive emf that occurs in it.
Magnetic field energy is equal to the work that the current does to overcome the self-induced emf during the time while the current increases from zero to the maximum value.
.
Electromagnetic vibrations– these are periodic changes in charge, current strength and all characteristics of electric and magnetic fields.
Electrical oscillatory system(oscillating circuit) consists of a capacitor and an inductor.
Conditions for the occurrence of oscillations:
The system must be brought out of equilibrium; for this, a charge is given to the capacitor. Electric field energy of a charged capacitor:
.
The system must return to a state of equilibrium. Under the influence of an electric field, charge transfers from one plate of the capacitor to another, that is, an electric current appears in the circuit, which flows through the coil. As the current increases in the inductor, a self-induction emf arises; the self-induction current is directed in the opposite direction. When the current in the coil decreases, the self-induction current is directed in the same direction. Thus, the self-induction current tends to return the system to a state of equilibrium.
The electrical resistance of the circuit should be low.
Ideal oscillatory circuit has no resistance. The vibrations in it are called free.
For any electrical circuit, Ohm's law is satisfied, according to which the emf acting in the circuit is equal to the sum of the voltages in all sections of the circuit. There is no current source in the oscillatory circuit, but a self-inductive emf appears in the inductor, which is equal to the voltage across the capacitor.
Conclusion: the charge of the capacitor changes according to a harmonic law.
Capacitor voltage:
.
Current strength in the circuit:
.
Magnitude 
- current amplitude.
The difference from the charge on 
.
Period of free oscillations in the circuit:
Electric field energy of a capacitor:
Coil magnetic field energy:
The energies of the electric and magnetic fields vary according to a harmonic law, but the phases of their oscillations are different: when the energy of the electric field is maximum, the energy of the magnetic field is zero.
Total energy of the oscillatory system:
.
IN ideal contour the total energy does not change.
During the oscillation process, the energy of the electric field is completely converted into the energy of the magnetic field and vice versa. This means that the energy at any moment in time is equal to either the maximum energy of the electric field or the maximum energy of the magnetic field.
Real oscillating circuit contains resistance. The vibrations in it are called fading.
Ohm's law will take the form:
Provided that the damping is small (the square of the natural frequency of oscillations is much greater than the square of the damping coefficient), the logarithmic damping decrement is:
With strong damping (the square of the natural frequency of oscillation is less than the square of the oscillation coefficient):
This equation describes the process of discharging a capacitor into a resistor. In the absence of inductance, oscillations will not occur. According to this law, the voltage on the capacitor plates also changes.
Total Energy in a real circuit decreases, since heat is released into the resistance R during the passage of current.
Transition process– a process that occurs in electrical circuits during the transition from one operating mode to another. Estimated by time ( 
), during which the parameter characterizing the transition process will change by e times.
For circuit with capacitor and resistor:
.
Maxwell's theory of the electromagnetic field:
1 position:
Any alternating electric field generates a vortex magnetic field. An alternating electric field was called a displacement current by Maxwell, since it, like an ordinary current, causes a magnetic field.
To detect the displacement current, consider the passage of current through a system in which a capacitor with a dielectric is connected.
Bias current density:
. The current density is directed in the direction of the voltage change.
Maxwell's first equation:
- the vortex magnetic field is generated by both conduction currents (moving electric charges) and displacement currents (alternating electric field E).
2 position:
Any alternating magnetic field generates a vortex electric field - the basic law of electromagnetic induction.
Maxwell's second equation:
- connects the rate of change of magnetic flux through any surface and the circulation of the electric field strength vector that arises at the same time.
Any conductor carrying current creates a magnetic field in space. If the current is constant (does not change over time), then the magnetic field associated with it is also constant. A changing current creates a changing magnetic field. There is an electric field inside a conductor carrying current. Therefore, a changing electric field creates a changing magnetic field.
The magnetic field is vortex, since the lines of magnetic induction are always closed. The magnitude of the magnetic field strength H is proportional to the rate of change of the electric field strength 
. Direction of the magnetic field strength vector 
associated with changes in electric field strength 
right screw rule: clench your right hand into a fist, point your thumb in the direction of the change in electric field strength, then the bent 4 fingers will indicate the direction of the magnetic field strength lines.
Any changing magnetic field creates a vortex electric field, whose tension lines are closed and located in a plane perpendicular to the magnetic field strength.
The magnitude of the intensity E of the vortex electric field depends on the rate of change of the magnetic field 
. The direction of vector E is related to the direction of change in the magnetic field H by the left screw rule: clench your left hand into a fist, point your thumb in the direction of the change in the magnetic field, bent four fingers will indicate the direction of the lines of intensity of the vortex electric field.
The set of interconnected vortex electric and magnetic fields represents electromagnetic field. The electromagnetic field does not remain at the place of origin, but propagates in space in the form of a transverse electromagnetic wave.
Electromagnetic wave– this is the propagation in space of vortex electric and magnetic fields associated with each other.
Condition for the occurrence of an electromagnetic wave– movement of the charge with acceleration.
Electromagnetic Wave Equation:
- cyclic frequency of electromagnetic oscillations
t– time from the beginning of oscillations
l – distance from the wave source to a given point in space
- wave propagation speed
The time it takes a wave to travel from its source to a given point.
Vectors E and H in an electromagnetic wave are perpendicular to each other and to the speed of propagation of the wave.
Source of electromagnetic waves– conductors through which rapidly alternating currents flow (macroemitters), as well as excited atoms and molecules (microemitters). The higher the oscillation frequency, the better electromagnetic waves are emitted in space.
Properties of electromagnetic waves:
All electromagnetic waves are transverse
In a homogeneous medium, electromagnetic waves propagate at a constant speed, which depends on the properties of the environment:
- relative dielectric constant of the medium
- dielectric constant of vacuum, 
F/m, Cl 2 /nm 2
- relative magnetic permeability of the medium
- magnetic constant of vacuum, 
N/A 2 ; Gn/m
Electromagnetic waves reflected from obstacles, absorbed, scattered, refracted, polarized, diffracted, interfered.
Volumetric energy density The electromagnetic field consists of the volumetric energy densities of the electric and magnetic fields:
Wave energy flux density - wave intensity:
-Umov-Poynting vector.
All electromagnetic waves are arranged in a series of frequencies or wavelengths ( 
). This row is electromagnetic wave scale.
Low frequency vibrations. 0 – 10 4 Hz. Obtained from generators. They radiate poorly
Radio waves. 10 4 – 10 13 Hz. They are emitted by solid conductors carrying rapidly alternating currents.
Infrared radiation– waves emitted by all bodies at temperatures above 0 K, due to intra-atomic and intra-molecular processes.
Visible light– waves that act on the eye, causing visual sensation. 380-760 nm
Ultraviolet radiation. 10 – 380 nm. Visible light and UV arise when the movement of electrons in the outer shells of an atom changes.
X-ray radiation. 80 – 10 -5 nm. Occurs when the movement of electrons in the inner shells of an atom changes.
Gamma radiation. Occurs during the decay of atomic nuclei.
Determination of magnetic field. His sources
Definition
A magnetic field is one of the forms of an electromagnetic field that acts only on moving bodies that have an electric charge or magnetized bodies, regardless of their movement.
The sources of this field are constant electric currents, moving electric charges (bodies and particles), magnetized bodies, alternating electric fields. The sources of constant magnetic field are direct currents.
Properties of magnetic field
At a time when the study of magnetic phenomena had just begun, researchers paid special attention to the fact that there are poles in magnetized bars. In them, the magnetic properties manifested themselves especially clearly. At the same time, it was clearly visible that the poles of the magnet were different. Opposite poles attracted, and like poles repelled. Gilbert proposed the idea of the existence of “magnetic charges”. These ideas were supported and developed by Coulomb. Based on Coulomb's experiments, the force characteristic of a magnetic field became the force with which the magnetic field acts on a magnetic charge equal to unity. Coulomb drew attention to the significant differences between the phenomena of electricity and magnetism. The difference is already evident in the fact that electric charges can be separated and get bodies with an excess of positive or negative charge, while it is impossible to separate the north and south poles of a magnet and get a body with only one pole. From the impossibility of dividing a magnet into exclusively “northern” or “southern”, Coulomb decided that these two types of charges are inseparable in each elementary particle of the magnetizing substance. Thus, it was recognized that every particle of matter - an atom, a molecule or a group of them - is something like a micro magnet with two poles. In this case, the magnetization of a body is the process of orientation of its elementary magnets under the influence of an external magnetic field (analogous to the polarization of dielectrics).
The interaction of currents is realized through magnetic fields. Oersted discovered that the magnetic field is excited by current and has an orienting effect on the magnetic needle. Oersted had a current-carrying conductor located above a magnetic needle, which could rotate. When current flowed in the conductor, the arrow turned perpendicular to the wire. A change in the direction of the current caused a reorientation of the needle. From Oersted's experiment it followed that the magnetic field has a direction and should be characterized by a vector quantity. This quantity was called magnetic induction and denoted: $\overrightarrow(B).$ $\overrightarrow(B)$ is similar to the strength vector for the electric field ($\overrightarrow(E)$). The analogue of the displacement vector $\overrightarrow(D)\ $for the magnetic field has become the vector $\overrightarrow(H)$ - called the magnetic field strength vector.
A magnetic field only affects a moving electric charge. A magnetic field is generated by moving electric charges.
Magnetic field of a moving charge. Magnetic field of a coil with current. Superposition principle
The magnetic field of an electric charge that moves at a constant speed has the form:
\[\overrightarrow(B)=\frac((\mu )_0)(4\pi )\frac(q\left[\overrightarrow(v)\overrightarrow(r)\right])(r^3)\left (1\right),\]
where $(\mu )_0=4\pi \cdot (10)^(-7)\frac(H)(m)(in\SI)$ is the magnetic constant, $\overrightarrow(v)$ is the speed movement of the charge, $\overrightarrow(r)$ is the radius vector that determines the location of the charge, q is the magnitude of the charge, $\left[\overrightarrow(v)\overrightarrow(r)\right]$ is the vector product.
Magnetic induction of an element with current in the SI system:
where $\ \overrightarrow(r)$ is the radius vector drawn from the current element to the point under consideration, $\overrightarrow(dl)$ is the element of the conductor with current (the direction of the current is specified), $\vartheta$ is the angle between $ \overrightarrow(dl)$ and $\overrightarrow(r)$. The direction of the vector $\overrightarrow(dB)$ is perpendicular to the plane in which $\overrightarrow(dl)$ and $\overrightarrow(r)$ lie. Determined by the right screw rule.
For a magnetic field, the superposition principle holds:
\[\overrightarrow(B)=\sum((\overrightarrow(B))_i\left(3\right),)\]
where $(\overrightarrow(B))_i$ are individual fields that are generated by moving charges, $\overrightarrow(B)$ is the total magnetic field induction.
Example 1
Task: Find the ratio of the forces of magnetic and Coulomb interaction of two electrons that move with the same speeds $v$ in parallel. The distance between particles is constant.
\[\overrightarrow(F_m)=q\left[\overrightarrow(v)\overrightarrow(B)\right]\left(1.1\right).\]
The field that creates the second moving electron is equal to:
\[\overrightarrow(B)=\frac((\mu )_0)(4\pi )\frac(q\left[\overrightarrow(v)\overrightarrow(r)\right])(r^3)\left (1.2\right).\]
Let the distance between electrons be equal to $a=r\ (constant)$. We use the algebraic property of the vector product (Lagrange’s identity ($\left[\overrightarrow(a)\left[\overrightarrow(b)\overrightarrow(c)\right]\right]=\overrightarrow(b)\left(\overrightarrow(a )\overrightarrow(c)\right)-\overrightarrow(c)\left(\overrightarrow(a)\overrightarrow(b)\right)$))
\[(\overrightarrow(F))_m=\frac((\mu )_0)(4\pi )\frac(q^2)(a^3)\left[\overrightarrow(v)\left[\overrightarrow (v)\overrightarrow(a)\right]\right]=\left(\overrightarrow(v)\left(\overrightarrow(v)\overrightarrow(a)\right)-\overrightarrow(a)\left(\overrightarrow (v)\overrightarrow(v)\right)\right)=-\frac((\mu )_0)(4\pi )\frac(q^2\overrightarrow(a)v^2)(a^3) \ ,\]
$\overrightarrow(v)\left(\overrightarrow(v)\overrightarrow(a)\right)=0$, since $\overrightarrow(v\bot )\overrightarrow(a)$.
Force modulus $F_m=\frac((\mu )_0)(4\pi )\frac(q^2v^2)(a^2),\ $where $q=q_e=1.6\cdot 10^( -19)Kl$.
The modulus of the Coulomb force, which acts on an electron, in the field is equal to:
Let's find the force ratio $\frac(F_m)(F_q)$:
\[\frac(F_m)(F_q)=\frac((\mu )_0)(4\pi )\frac(q^2v^2)(a^2):\frac(q^2)((4 \pi (\varepsilon )_0a)^2)=(\mu )_0((\varepsilon )_0v)^2.\]
Answer: $\frac(F_m)(F_q)=(\mu )_0((\varepsilon )_0v)^2.$
Example 2
Task: A direct current of force I circulates along a coil with current in the form of a circle of radius R. Find the magnetic induction in the center of the circle.
Let us select an elementary section on the current-carrying conductor (Fig. 1); as a basis for solving the problem, we use the induction formula for a current-carrying coil element:
where $\ \overrightarrow(r)$ is the radius vector drawn from the current element to the point under consideration, $\overrightarrow(dl)$ is the element of the conductor with current (the direction of the current is specified), $\vartheta$ is the angle between $ \overrightarrow(dl)$ and $\overrightarrow(r)$. Based on Fig. 1 $\vartheta=90()^\circ $, therefore (2.1) will be simplified, in addition, the distance from the center of the circle (the point where we are looking for the magnetic field) of the conductor element with current is constant and equal to the radius of the turn (R), therefore we have:
All current elements will generate magnetic fields that are directed along the x axis. This means that the resulting magnetic field induction vector can be found as the sum of the projections of individual vectors$\ \ \overrightarrow(dB).$ Then, according to the principle of superposition, the total magnetic field induction can be obtained by passing to the integral:
Substituting (2.2) into (2.3), we get:
Answer: $B$=$\frac((\mu )_0)(2)\frac(I)(R).$
Magnetic field- this is the material medium through which interaction occurs between conductors with current or moving charges.
Properties of magnetic field:
Characteristics of the magnetic field:
To study the magnetic field, a test circuit with current is used. It is small in size, and the current in it is much less than the current in the conductor creating the magnetic field. On opposite sides of the current-carrying circuit, forces from the magnetic field act that are equal in magnitude, but directed in opposite directions, since the direction of the force depends on the direction of the current. The points of application of these forces do not lie on the same straight line. Such forces are called a couple of forces. As a result of the action of a pair of forces, the circuit cannot move translationally; it rotates around its axis. The rotating action is characterized torque.
, Where l–leverage couple of forces(distance between points of application of forces).
As the current in the test circuit or the area of the circuit increases, the torque of the pair of forces will increase proportionally. The ratio of the maximum moment of force acting on the circuit with current to the magnitude of the current in the circuit and the area of the circuit is a constant value for a given point in the field. It's called magnetic induction.
, Where 
-magnetic moment circuit with current.
Unit of measurement magnetic induction – Tesla [T].
Magnetic moment of the circuit– vector quantity, the direction of which depends on the direction of the current in the circuit and is determined by right screw rule: clench your right hand into a fist, point four fingers in the direction of the current in the circuit, then the thumb will indicate the direction of the magnetic moment vector. The magnetic moment vector is always perpendicular to the contour plane.
For direction of the magnetic induction vector take the direction of the vector of the magnetic moment of the circuit, oriented in the magnetic field.
Magnetic induction line– a line whose tangent at each point coincides with the direction of the magnetic induction vector. Magnetic induction lines are always closed and never intersect. Magnetic induction lines of a straight conductor with current have the form of circles located in a plane perpendicular to the conductor. The direction of the magnetic induction lines is determined by the right-hand screw rule. Magnetic induction lines of circular current(turns with current) also have the form of circles. Each coil element is length 
can be imagined as a straight conductor that creates its own magnetic field. For magnetic fields, the principle of superposition (independent addition) applies. The total vector of magnetic induction of the circular current is determined as the result of the addition of these fields in the center of the turn according to the right-hand screw rule.
If the magnitude and direction of the magnetic induction vector are the same at every point in space, then the magnetic field is called homogeneous. If the magnitude and direction of the magnetic induction vector at each point do not change over time, then such a field is called permanent.
Magnitude magnetic induction at any point in the field is directly proportional to the current strength in the conductor creating the field, inversely proportional to the distance from the conductor to a given point in the field, depends on the properties of the medium and the shape of the conductor creating the field.
, Where 
N/A 2 ; Gn/m – magnetic constant of vacuum,
-relative magnetic permeability of the medium,
-absolute magnetic permeability of the medium.
Depending on the value of magnetic permeability, all substances are divided into three classes:
As the absolute permeability of the medium increases, the magnetic induction at a given point in the field also increases. The ratio of magnetic induction to the absolute magnetic permeability of the medium is a constant value for a given poly point, e is called tension.
.
The vectors of tension and magnetic induction coincide in direction. The magnetic field strength does not depend on the properties of the medium.
Ampere power– the force with which the magnetic field acts on a current-carrying conductor.
Where l– length of the conductor, 
- the angle between the magnetic induction vector and the direction of the current.
The direction of the Ampere force is determined by left hand rule: the left hand is positioned so that the component of the magnetic induction vector, perpendicular to the conductor, enters the palm, four extended fingers are directed along the current, then the thumb bent by 90 0 will indicate the direction of the Ampere force.
The result of the Ampere force is the movement of the conductor in a given direction.
E 
if 
= 90 0 , then F=max, if 
= 0 0 , then F = 0.
Lorentz force– the force of the magnetic field on a moving charge.
, where q is the charge, v is the speed of its movement, 
- the angle between the vectors of tension and speed.
The Lorentz force is always perpendicular to the magnetic induction and velocity vectors. The direction is determined by left hand rule(fingers follow the movement of the positive charge). If the direction of the particle's velocity is perpendicular to the magnetic induction lines of a uniform magnetic field, then the particle moves in a circle without changing its kinetic energy.
Since the direction of the Lorentz force depends on the sign of the charge, it is used to separate charges.
Magnetic flux– a value equal to the number of magnetic induction lines that pass through any area located perpendicular to the magnetic induction lines.
, Where 
- the angle between the magnetic induction and the normal (perpendicular) to the area S.
Unit of measurement– Weber [Wb].
Magnetic flux measurement methods:
Changing the orientation of the site in a magnetic field (changing the angle)
Changing the area of a circuit placed in a magnetic field
Change in current strength creating a magnetic field
Changing the distance of the circuit from the magnetic field source
Changes in the magnetic properties of the medium.
F 
Araday recorded an electric current in a circuit that did not contain a source, but was located next to another circuit containing a source. Moreover, the current in the first circuit arose in the following cases: with any change in the current in circuit A, with relative movement of the circuits, with the introduction of an iron rod into circuit A, with the movement of a permanent magnet relative to circuit B. Directed movement of free charges (current) occurs only in an electric field. This means that a changing magnetic field generates an electric field, which sets in motion the free charges of the conductor. This electric field is called induced or vortex.
Differences between a vortex electric field and an electrostatic one:
The source of the vortex field is a changing magnetic field.
The vortex field intensity lines are closed.
The work done by this field to move a charge along a closed circuit is not zero.
The energy characteristic of a vortex field is not the potential, but induced emf– a value equal to the work of external forces (forces of non-electrostatic origin) to move a unit of charge along a closed circuit.
.Measured in Volts[IN].
A vortex electric field occurs with any change in the magnetic field, regardless of whether there is a conducting closed circuit or not. The circuit only allows one to detect the vortex electric field.
Electromagnetic induction- this is the occurrence of induced emf in a closed circuit with any change in the magnetic flux through its surface.
The induced emf in a closed circuit generates an induced current.
.
Direction of induction current determined by Lenz's rule: the induced current is in such a direction that the magnetic field created by it counteracts any change in the magnetic flux that generated this current.
Faraday's law for electromagnetic induction: The induced emf in a closed loop is directly proportional to the rate of change of magnetic flux through the surface bounded by the loop.
T 
oki fuko– eddy induction currents that arise in large conductors placed in a changing magnetic field. The resistance of such a conductor is low, since it has a large cross-section S, so the Foucault currents can be large in value, as a result of which the conductor heats up.
Self-induction- this is the occurrence of induced emf in a conductor when the current strength in it changes.
A conductor carrying current creates a magnetic field. Magnetic induction depends on the current strength, therefore the intrinsic magnetic flux also depends on the current strength.
, where L is the proportionality coefficient, inductance.
Unit of measurement inductance – Henry [H].
Inductance conductor depends on its size, shape and magnetic permeability of the medium.
Inductance increases with increasing length of the conductor, the inductance of a turn is greater than the inductance of a straight conductor of the same length, the inductance of a coil (a conductor with a large number of turns) is greater than the inductance of one turn, the inductance of a coil increases if an iron rod is inserted into it.
Faraday's law for self-induction:
.
Self-induced emf is directly proportional to the rate of change of current.
Self-induced emf generates a self-induction current, which always prevents any change in the current in the circuit, that is, if the current increases, the self-induction current is directed in the opposite direction; when the current in the circuit decreases, the self-induction current is directed in the same direction. The greater the inductance of the coil, the greater the self-inductive emf that occurs in it.
Magnetic field energy is equal to the work that the current does to overcome the self-induced emf during the time while the current increases from zero to the maximum value.
.
Electromagnetic vibrations– these are periodic changes in charge, current strength and all characteristics of electric and magnetic fields.
Electrical oscillatory system(oscillating circuit) consists of a capacitor and an inductor.
Conditions for the occurrence of oscillations:
The system must be brought out of equilibrium; for this, a charge is given to the capacitor. Electric field energy of a charged capacitor:
.
The system must return to a state of equilibrium. Under the influence of an electric field, charge transfers from one plate of the capacitor to another, that is, an electric current appears in the circuit, which flows through the coil. As the current increases in the inductor, a self-induction emf arises; the self-induction current is directed in the opposite direction. When the current in the coil decreases, the self-induction current is directed in the same direction. Thus, the self-induction current tends to return the system to a state of equilibrium.
The electrical resistance of the circuit should be low.
Ideal oscillatory circuit has no resistance. The vibrations in it are called free.
For any electrical circuit, Ohm's law is satisfied, according to which the emf acting in the circuit is equal to the sum of the voltages in all sections of the circuit. There is no current source in the oscillatory circuit, but a self-inductive emf appears in the inductor, which is equal to the voltage across the capacitor.
Conclusion: the charge of the capacitor changes according to a harmonic law.
Capacitor voltage:
.
Current strength in the circuit:
.
Magnitude 
- current amplitude.
The difference from the charge on 
.
Period of free oscillations in the circuit:
Electric field energy of a capacitor:
Coil magnetic field energy:
The energies of the electric and magnetic fields vary according to a harmonic law, but the phases of their oscillations are different: when the energy of the electric field is maximum, the energy of the magnetic field is zero.
Total energy of the oscillatory system:
.
IN ideal contour the total energy does not change.
During the oscillation process, the energy of the electric field is completely converted into the energy of the magnetic field and vice versa. This means that the energy at any moment in time is equal to either the maximum energy of the electric field or the maximum energy of the magnetic field.
Real oscillating circuit contains resistance. The vibrations in it are called fading.
Ohm's law will take the form:
Provided that the damping is small (the square of the natural frequency of oscillations is much greater than the square of the damping coefficient), the logarithmic damping decrement is:
With strong damping (the square of the natural frequency of oscillation is less than the square of the oscillation coefficient):
This equation describes the process of discharging a capacitor into a resistor. In the absence of inductance, oscillations will not occur. According to this law, the voltage on the capacitor plates also changes.
Total Energy in a real circuit decreases, since heat is released into the resistance R during the passage of current.
Transition process– a process that occurs in electrical circuits during the transition from one operating mode to another. Estimated by time ( 
), during which the parameter characterizing the transition process will change by e times.
For circuit with capacitor and resistor:
.
Maxwell's theory of the electromagnetic field:
1 position:
Any alternating electric field generates a vortex magnetic field. An alternating electric field was called a displacement current by Maxwell, since it, like an ordinary current, causes a magnetic field.
To detect the displacement current, consider the passage of current through a system in which a capacitor with a dielectric is connected.
Bias current density:
. The current density is directed in the direction of the voltage change.
Maxwell's first equation:
- the vortex magnetic field is generated by both conduction currents (moving electric charges) and displacement currents (alternating electric field E).
2 position:
Any alternating magnetic field generates a vortex electric field - the basic law of electromagnetic induction.
Maxwell's second equation:
- connects the rate of change of magnetic flux through any surface and the circulation of the electric field strength vector that arises at the same time.
Any conductor carrying current creates a magnetic field in space. If the current is constant (does not change over time), then the magnetic field associated with it is also constant. A changing current creates a changing magnetic field. There is an electric field inside a conductor carrying current. Therefore, a changing electric field creates a changing magnetic field.
The magnetic field is vortex, since the lines of magnetic induction are always closed. The magnitude of the magnetic field strength H is proportional to the rate of change of the electric field strength 
. Direction of the magnetic field strength vector 
associated with changes in electric field strength 
right screw rule: clench your right hand into a fist, point your thumb in the direction of the change in electric field strength, then the bent 4 fingers will indicate the direction of the magnetic field strength lines.
Any changing magnetic field creates a vortex electric field, whose tension lines are closed and located in a plane perpendicular to the magnetic field strength.
The magnitude of the intensity E of the vortex electric field depends on the rate of change of the magnetic field 
. The direction of vector E is related to the direction of change in the magnetic field H by the left screw rule: clench your left hand into a fist, point your thumb in the direction of the change in the magnetic field, bent four fingers will indicate the direction of the lines of intensity of the vortex electric field.
The set of interconnected vortex electric and magnetic fields represents electromagnetic field. The electromagnetic field does not remain at the place of origin, but propagates in space in the form of a transverse electromagnetic wave.
Electromagnetic wave– this is the propagation in space of vortex electric and magnetic fields associated with each other.
Condition for the occurrence of an electromagnetic wave– movement of the charge with acceleration.
Electromagnetic Wave Equation:
- cyclic frequency of electromagnetic oscillations
t– time from the beginning of oscillations
l – distance from the wave source to a given point in space
- wave propagation speed
The time it takes a wave to travel from its source to a given point.
Vectors E and H in an electromagnetic wave are perpendicular to each other and to the speed of propagation of the wave.
Source of electromagnetic waves– conductors through which rapidly alternating currents flow (macroemitters), as well as excited atoms and molecules (microemitters). The higher the oscillation frequency, the better electromagnetic waves are emitted in space.
Properties of electromagnetic waves:
All electromagnetic waves are transverse
In a homogeneous medium, electromagnetic waves propagate at a constant speed, which depends on the properties of the environment:
- relative dielectric constant of the medium
- dielectric constant of vacuum, 
F/m, Cl 2 /nm 2
- relative magnetic permeability of the medium
- magnetic constant of vacuum, 
N/A 2 ; Gn/m
Electromagnetic waves reflected from obstacles, absorbed, scattered, refracted, polarized, diffracted, interfered.
Volumetric energy density The electromagnetic field consists of the volumetric energy densities of the electric and magnetic fields:
Wave energy flux density - wave intensity:
-Umov-Poynting vector.
All electromagnetic waves are arranged in a series of frequencies or wavelengths ( 
). This row is electromagnetic wave scale.
Low frequency vibrations. 0 – 10 4 Hz. Obtained from generators. They radiate poorly
Radio waves. 10 4 – 10 13 Hz. They are emitted by solid conductors carrying rapidly alternating currents.
Infrared radiation– waves emitted by all bodies at temperatures above 0 K, due to intra-atomic and intra-molecular processes.
Visible light– waves that act on the eye, causing visual sensation. 380-760 nm
Ultraviolet radiation. 10 – 380 nm. Visible light and UV arise when the movement of electrons in the outer shells of an atom changes.
X-ray radiation. 80 – 10 -5 nm. Occurs when the movement of electrons in the inner shells of an atom changes.
Gamma radiation. Occurs during the decay of atomic nuclei.
The term “magnetic field” usually means a certain energy space in which the forces of magnetic interaction manifest themselves. They affect:
individual substances: ferrimagnets (metals - mainly cast iron, iron and their alloys) and their class of ferrites, regardless of state;
moving charges of electricity.
Physical bodies that have a total magnetic moment of electrons or other particles are called permanent magnets. Their interaction is shown in the picture magnetic power lines.
They were formed after bringing a permanent magnet to the back of a cardboard sheet with an even layer of iron filings. The picture shows clear markings of the north (N) and south (S) poles with the direction of the field lines relative to their orientation: exit from the north pole and entrance to the south.
How is a magnetic field created?
The sources of the magnetic field are:
permanent magnets;
moving charges;
time-varying electric field.
Every kindergarten child is familiar with the action of permanent magnets. After all, he already had to sculpt pictures of magnets on the refrigerator, taken from packages with all sorts of delicacies.
Electric charges in motion usually have significantly greater magnetic field energy than . It is also designated by lines of force. Let's look at the rules for drawing them for a straight conductor with current I.
The magnetic field line is drawn in a plane perpendicular to the movement of the current so that at each point the force acting on the north pole of the magnetic needle is directed tangentially to this line. This creates concentric circles around the moving charge.
The direction of these forces is determined by the well-known rule of a screw or gimlet with right-hand thread winding.
Gimlet rule
It is necessary to position the gimlet coaxially with the current vector and rotate the handle so that the translational movement of the gimlet coincides with its direction. Then the orientation of the magnetic field lines will be shown by rotating the handle.
In a ring conductor, the rotational movement of the handle coincides with the direction of the current, and the translational movement indicates the orientation of the induction.
Magnetic lines of force always leave the north pole and enter the south pole. They continue inside the magnet and are never open.
Rules for the interaction of magnetic fields
Magnetic fields from different sources add to each other to form a resulting field.
In this case, magnets with opposite poles (N - S) attract each other, and with like poles (N - N, S - S) they repel. The interaction forces between the poles depend on the distance between them. The closer the poles are shifted, the greater the force generated.
Basic characteristics of the magnetic field
These include:
magnetic induction vector (B);
magnetic flux (F);
flux linkage (Ψ).
The intensity or strength of the field impact is estimated by the value magnetic induction vector. It is determined by the value of the force “F” created by the passing current “I” through a conductor of length “l”. В =F/(I∙l)
The unit of measurement of magnetic induction in the SI system is Tesla (in memory of the physicist who studied these phenomena and described them using mathematical methods). In Russian technical literature it is designated “Tl”, and in international documentation the symbol “T” is adopted.
1 T is the induction of such a uniform magnetic flux, which acts with a force of 1 newton for each meter of length of a straight conductor perpendicular to the direction of the field, when a current of 1 ampere passes through this conductor.
1T=1∙N/(A∙m)
The direction of vector B is determined by left hand rule.
If you place the palm of your left hand in a magnetic field so that the lines of force from the north pole enter the palm at a right angle, and place four fingers in the direction of the current in the conductor, then the protruding thumb will indicate the direction of the force on this conductor.
In the case when the conductor with electric current is not located at right angles to the magnetic lines of force, the force acting on it will be proportional to the magnitude of the flowing current and the component of the projection of the length of the conductor with current onto a plane located in the perpendicular direction.
The force acting on an electric current does not depend on the materials from which the conductor is made and its cross-sectional area. Even if this conductor does not exist at all, and moving charges begin to move in another medium between the magnetic poles, then this force will not change in any way.
If inside a magnetic field at all points the vector B has the same direction and magnitude, then such a field is considered uniform.
Any environment that has , affects the value of the induction vector B .
Magnetic flux (F)
If we consider the passage of magnetic induction through a certain area S, then the induction limited by its limits will be called magnetic flux.
When the area is inclined at some angle α to the direction of magnetic induction, the magnetic flux decreases by the amount of the cosine of the angle of inclination of the area. Its maximum value is created when the area is perpendicular to its penetrating induction. Ф=В·S
The unit of measurement for magnetic flux is 1 weber, defined by the passage of induction of 1 tesla through an area of 1 square meter.
Flux linkage
This term is used to obtain the total amount of magnetic flux created from a certain number of current-carrying conductors located between the poles of a magnet.
For the case when the same current I passes through the winding of a coil with a number of turns n, then the total (linked) magnetic flux from all turns is called flux linkage Ψ.
Ψ=n·Ф . The unit of flux linkage is 1 weber.
How is a magnetic field formed from an alternating electric
The electromagnetic field, interacting with electric charges and bodies with magnetic moments, is a combination of two fields:
electrical;
magnetic.
They are interconnected, represent a combination of each other, and when one changes over time, certain deviations occur in the other. For example, when an alternating sinusoidal electric field is created in a three-phase generator, the same magnetic field with the characteristics of similar alternating harmonics is simultaneously formed.
Magnetic properties of substances
In relation to interaction with an external magnetic field, substances are divided into:
antiferromagnets with balanced magnetic moments, due to which a very low degree of magnetization of the body is created;
Diamagnets with the property of magnetizing an internal field against the action of an external one. When there is no external field, their magnetic properties do not appear;
paramagnetic materials with magnetizing properties of the internal field in the direction of the external field, which have a low degree;
ferromagnets, which have magnetic properties without an applied external field at temperatures below the Curie point;
ferrimagnets with magnetic moments unbalanced in magnitude and direction.
All these properties of substances have found various applications in modern technology.
Magnetic circuits
All transformers, inductors, electrical machines and many other devices operate on this basis.
For example, in a working electromagnet, the magnetic flux passes through a magnetic core made of ferromagnetic steel and air with pronounced non-ferromagnetic properties. The combination of these elements makes up a magnetic circuit.
Most electrical devices have magnetic circuits in their design. Read more about this in this article -
