Inductance, or self-induction coefficient(from lat. indactio- guidance, excitation) - is a parameter of an electrical circuit that determines the self-inductive emf, which is induced in the circuit when the current flowing through it changes and/or its deformation.
The term “inductance” also refers to a self-induction coil, which determines the inductive properties of the circuit.
Self-induction- the formation of induced emf in a conducting circuit when the current strength changes in it. Self-induction was discovered in 1832 by the American scientist J. Henry. Independently of him, this phenomenon was discovered by M. Faraday in 1835.
An induced emf is formed when a magnetic flux changes. If this change is caused by its own current, then they speak of self-induced emf:
Where L- inductance of the circuit, or its self-induction coefficient.
Inductance, like electrical capacitance, depends on the geometry of the conductor - its size and shape, but is not dependent on the current strength in the conductor. Thus, the inductance of a straight wire is much less than the inductance of the same wire coiled.
Calculations show that the inductance of the solenoid described above in air is calculated by the formula:
.
Where μ 0 — magnetic constant, N- number of solenoid turns, l— solenoid length, S- cross-sectional area.
Also, inductance depends on the magnetic properties of the medium in which the conductor is located, namely on its magnetic permeability, which is determined using the formula:
Where L 0 - circuit inductance in vacuum, L- the inductance of a circuit in a homogeneous substance that fills a magnetic field.
The SI unit of inductance is Henry(H): 1 H = 1 V s/A.
Closing and opening currents.
Every time the current is turned on and off in the circuit, so-called extracurrents self-induction (extracurrents closures And erosion), which arise in the circuit due to the phenomenon of self-induction and which, according to Lenz’s rule, prevent the increase or decrease of current in the circuit.
The figure above shows a connection diagram for 2 identical lamps. One of them is connected to the source through a resistor R, and the other is connected in series with the coil L with an iron core. When the circuit is closed, the first lamp flashes almost instantly, and the second with a significant delay. This is due to the fact that the self-induction emf in the circuit of this lamp is large, and the current strength does not immediately reach its maximum value.
When the key in the coil is opened L A self-induced emf is formed, which maintains the initial current.
As a result, at the moment of opening, a current flows through the galvanometer (light arrow), which is directed opposite the initial current before opening (black arrow). In this case, the self-induction EMF can be much greater than the EMF of the battery of elements, which will manifest itself in the fact that the extra opening current will greatly exceed the stationary current when the switch is closed.
Inductance characterizes the inertia of a circuit in relation to changes in current in it, and it can be considered as an electrodynamic analogue of body mass in mechanics, which is a measure of the inertia of a body. In this case, the current I plays the role of body speed.
So far we have considered changing magnetic fields without paying attention to what their source is. In practice, magnetic fields are most often created using various types of solenoids, i.e. multi-turn circuits with current.
There are two possible cases here: when the current in the circuit changes, the magnetic flux changes: a ) the same circuit ; b ) adjacent circuit.
The induced emf arising in the circuit itself is called Self-induced emf, and the phenomenon itself – self-induction.
If the induced emf occurs in the adjacent circuit, then they talk about the phenomenon mutual induction.
It is clear that the nature of the phenomenon is the same, but different names are used to emphasize the place of occurrence of the induced emf.
Self-induction phenomenon discovered by the American scientist J. Henry.
| Henry Joseph(1797–1878) - American physicist, member of the National Academy of Sciences, its president (1866–1878). Works devoted to electromagnetism. The first designed powerful horseshoe-shaped electromagnets (1828), using multilayer windings of insulated wire (their load capacity reached one ton), and discovered the principle of electromagnetic induction in 1831 (M. Faraday was the first to publish the discovery of induction). He built an electric motor (1831), discovered (1832) the phenomenon of self-induction and extra-current, and established the causes affecting the inductance of a circuit. Invented the electromagnetic relay. He built a telegraph that operated on the territory of Princeton College, and in 1842 established the oscillatory nature of the capacitor discharge. |
The phenomenon of self-induction can be defined as follows.
Current I flowing in any circuit creates a magnetic flux F penetrating the same circuit. When I changes, F will change. Consequently, an induced emf will be induced in the circuit.
Because magnetic induction IN proportional to current I hence
Where L – proportionality coefficient, called circuit inductance .
If there are no ferromagnets inside the circuit, then
(because 
).
Loop inductance L depends on the geometry of the circuit, the number of turns, and the area of the circuit turn.
The SI unit of inductance is the inductance of a circuit in which a total flux occurs when current flows. This unit is called Henry (Gn).
Inductance dimension:
Let's calculate the inductance of the solenoid L . If the solenoid length l much larger than its diameter d ( ) , then the formulas for an infinitely long solenoid can be applied to it. Then
Here N – number of turns. Flow through each of the turns
Flux linkage
But we know that where the inductance of the solenoid comes from
Where n
– number of turns per unit length, i.e. 
is the volume of the solenoid, which means
| , | (5.1.1) |
From this formula you can find the dimension for the magnetic constant:
When the current changes in the circuit, a self-inductive emf arises equal to:
| , | (5.1.2) |
The minus sign in this formula is due to Lenz's rule.
The phenomenon of self-induction plays an important role in electrical and radio engineering. As we will see later, due to self-induction, the capacitor connected in series with the inductor is recharged, resulting in such L.C.-chain (oscillatory circuit) electromagnetic oscillations arise.
SELF-INDUCTION
Each conductor through which electricity flows. current is in its own magnetic field.
When the current strength changes in the conductor, the m.field changes, i.e. the magnetic flux created by this current changes. A change in magnetic flux leads to the emergence of a vortex electric. fields and an induced emf appears in the circuit. 
This phenomenon is called self-induction.
Self-induction is the phenomenon of the occurrence of induced emf in electricity. circuit as a result of changes in current strength.
The resulting emf is called Self-induced emf
Circuit closure 
When shorted in electrical circuit, the current increases, which causes an increase in the magnetic flux in the coil, and a vortex electric occurs. field directed against the current, i.e. A self-induction emf arises in the coil, preventing the increase in current in the circuit (the vortex field inhibits the electrons).
As a result L1 lights up later, than L2.
Open circuit 
When the electrical circuit is opened, the current decreases, a decrease in the flux in the coil occurs, and a vortex electrical field appears, directed like a current (trying to maintain the same current strength), i.e. A self-induced emf arises in the coil, maintaining the current in the circuit.
As a result, L when turned off flashes brightly.
Conclusion
in electrical engineering, the phenomenon of self-induction manifests itself when the circuit is closed (the electric current increases gradually) and when the circuit is opened (the electric current does not disappear immediately).
What does self-induced emf depend on?
Email the current creates its own magnetic field. The magnetic flux through the circuit is proportional to the magnetic field induction (Ф ~ B), the induction is proportional to the current strength in the conductor
(B ~ I), therefore the magnetic flux is proportional to the current strength (Ф ~ I).
The self-induction emf depends on the rate of change of current in the electric current. circuit, from the properties of the conductor
(size and shape) and on the relative magnetic permeability of the medium in which the conductor is located.
A physical quantity showing the dependence of the self-induction emf on the size and shape of the conductor and on the environment in which the conductor is located is called the self-induction coefficient or inductance. 
Inductance - physical. a value numerically equal to the self-inductive emf that occurs in the circuit when the current changes by 1 Ampere in 1 second.
Inductance can also be calculated using the formula:
where Ф is the magnetic flux through the circuit, I is the current strength in the circuit.
Units of inductance in the SI system:
The inductance of the coil depends on:
the number of turns, the size and shape of the coil and the relative magnetic permeability of the medium
(core possible).
The self-inductive emf prevents the current from increasing when the circuit is turned on and the current from decreasing when the circuit is opened.
Around a current-carrying conductor there is a magnetic field that has energy.
Where does it come from? The current source included in the electric chain has a reserve of energy.
At the moment of electrical closure. The current source circuit expends part of its energy to overcome the effect of the self-inductive emf that arises. This part of the energy, called the current’s own energy, goes to the formation of a magnetic field.
The magnetic field energy is own current energy.
The self-energy of the current is numerically equal to the work that the current source must do to overcome the self-induction emf in order to create a current in the circuit.
The energy of the magnetic field created by the current is directly proportional to the square of the current.
Where does the magnetic field energy go after the current stops? - stands out (when a circuit with a sufficiently large current is opened, a spark or arc may occur)
QUESTIONS FOR TEST PAPER
on the topic "Electromagnetic induction"
|
1. List 6 ways to obtain induction current. |
Inductance
Inductance
Current I, flowing in a closed loop, creates a magnetic field around itself B .
F ~ I.
where is the proportionality coefficient L called circuit inductance .
Self-induction phenomenon
When the current changes I the magnetic field it creates changes in the circuit. Consequently, an emf is induced in the circuit.
This process is called self-induction .
In the SI system, inductance is measured in henry: [ L] = Gn = Vb/A = V s/A.
Self-induction phenomenon
E.m.f. induction E i created by an external magnetic field.
E.m.f. self-induction E S is created when its own magnetic field changes.
In general, the loop inductance L depends on
1) the geometric shape of the contour and its dimensions,
2) magnetic permeability of the medium in which the circuit is located.
In electrostatics, an analogue of inductance is electrical capacitance WITH solitary conductor, which depends on the shape, size, dielectric constant ε environment.
L = const, if magnetic permeability μ the environment and the geometric dimensions of the contour are constant.
Faraday's law for self-induction
The minus sign in Faraday's law, in accordance with Lenz's rule, means that the presence of inductance L leads to a slower change in current I in the circuit.
If the current I increases, then dI/dt> 0 and, accordingly, E S < 0, т.е. ток самоиндукции IS directed towards the current I
If the current I increases, then dI/dt> 0 and, accordingly, E S < 0, т.е. ток самоиндукции IS directed towards the current I external source and slows down its growth.
If the current I decreases, then dI/dt< 0 и, соответственно, ES> 0, i.e. self-induction current IS has the same direction as the decreasing current I external source and slows down its decrease.
^ Faraday's law for self-induction
If the circuit has a certain inductance L, then any change in current I the more it slows down, the more L contour, i.e. the circuit has electrical inertia .
Solenoid inductance
Inductance L depends only on the geometric dimensions of the circuit and magnetic permeability μ environment.
ФN– flux of magnetic induction through N turns,
F = B.S.- magnetic flux through the pad S, limited to one turn.
Solenoid inductance
Solenoid field:
l– solenoid length,
n = N/ l– number of turns per unit length of the solenoid.
(2) (1):
According to Lenz's rule, when turning on and off the current in a circuit containing inductance L, a self-induction current occurs IS, which is directed so as to prevent the current from changing I in the chain.
Extra currents
Key TO pregnant 1 :
Key TO pregnant 2 (open circuit):
E arises S and the current caused by it
Extra currents
constant called relaxation time – time during which the current strength I decreases in e once.
The more L, the more τ , and the slower the current decreases I.
Extra currents
At circuit closure in addition to the external emf. E emf arises. self-induction E S.
Extra currents
At the moment of closure t= 0 current I= 0, variable a 0 = – I 0, at time t current strength I, variable a =I – I 0
Extra currents
I 0 – steady current.
The establishment of current occurs the faster, the smaller L circuit and its greater resistance R
Extra currents of closing and breaking
Because battery resistance r is usually small, then we can assume that R R 0, where
R 0 – circuit resistance without taking into account the resistance of the EMF source. Steady current
R 0 to R.
● Instant increase in circuit resistance from R 0 to R.
The steady current was
At turning off the source e.m.f.
(open circuit) the current varies according to the law
The magnitude of the e.m.f. self-induction
RR>>R 0), then E S
If the circuit switches against very high external resistance R, for example, the chain breaks ( R>>R 0), then E S can become huge and a voltaic arc is formed between the open ends of the switch.
e.m.f. self-induction
In a circuit with high inductance, E S there may be more emf. source E included in the circuit, which can lead to insulation breakdown and equipment failure.
Therefore, resistance must be introduced into the circuit gradually, reducing the ratio dI /dt.
Mutual induction
The magnetic flux formed by circuit 1 penetrates circuit 2:
L 21 – proportionality coefficient.
If I 1 changes, then an emf is induced in circuit 2.
Mutual induction
Similarly, if circuit 2 changes I 2, then in the first circuit a change in the magnetic flux induces an emf:
Odds L 12 = L 21 – mutual inductance contours depends on
1. geometric shape,
2. sizes,
3. mutual position,
4. magnetic permeability of the medium μ .
For two coils on a common toroidal core
N 1, N 2 – number of turns of the first and second circuit, respectively,
l– length of the core (toroid) along the midline,
S– core section.
Transformer - a device consisting of two or more coils wound on one common core.
Serve to increase or decrease AC voltage:
transformation ratio.
Structurally, transformers are designed in such a way that the magnetic field is almost completely concentrated in the core.
In most transformers, the secondary winding is wound on top of the primary winding.
Autotransformer – a transformer consisting of one winding.
Boosting:
1-2 U supplied, 1-3 U removed.
Downgrade:
1-3 U supplied, 1-2 U removed.
Skin effect
When alternating current passes through a conductor, the magnetic field inside the conductor changes. A time-varying magnetic field generates in a conductor self-induction eddy currents .
Skin effect
The planes of eddy currents pass through the axis of the conductor.
According to Lenz's rule, eddy currents prevent the main current from changing inside the conductor and promote its change near the surface.
For alternating current, the resistance inside the conductor is greater than the resistance at the surface R inside > R on top
Skin effect
The alternating current density is not the same across the cross section:
jmax on a surface, jmin inside on the axis.
This phenomenon is called skin effect .
Consequence of the skin effect
RF currents flow through a thin surface layer, so the conductors for them are made hollow, and part of the outer surface is coated with silver.
Application:
a method of surface hardening of metals in which, when heated by high-frequency currents (HF), only the surface layer is heated.
Magnetic field energy. Volumetric magnetic field energy density
The energy of a magnetic field is equal to the work expended by the current to create this field.
Work due to induction phenomena
Magnetic field energy
Job dA is spent on changing the magnetic flux by the amount dФ.
Work to create magnetic flux F:
Volumetric magnetic field energy density
We'll find ω for example a solenoid
In this lesson, we will learn how and by whom the phenomenon of self-induction was discovered, consider the experience with which we will demonstrate this phenomenon, and determine that self-induction is a special case of electromagnetic induction. At the end of the lesson, we will introduce a physical quantity showing the dependence of the self-inductive emf on the size and shape of the conductor and on the environment in which the conductor is located, i.e. inductance.
Henry invented flat coils made of strip copper, with the help of which he achieved power effects that were more pronounced than when using wire solenoids. The scientist noticed that when there is a powerful coil in the circuit, the current in this circuit reaches its maximum value much more slowly than without the coil.
Rice. 2. Diagram of the experimental setup by D. Henry
In Fig. Figure 2 shows an electrical diagram of the experimental setup, on the basis of which the phenomenon of self-induction can be demonstrated. An electrical circuit consists of two parallel-connected light bulbs connected through a switch to a direct current source. A coil is connected in series with one of the light bulbs. After closing the circuit, it can be seen that the light bulb, which is connected in series with the coil, lights up more slowly than the second light bulb (Fig. 3).
Rice. 3. Different incandescence of light bulbs at the moment the circuit is turned on
When the source is turned off, the light bulb connected in series with the coil goes out more slowly than the second light bulb.
Why don't the lights go out at the same time?
When the switch is closed (Fig. 4), due to the occurrence of self-induction emf, the current in the light bulb with the coil increases more slowly, so this light bulb lights up more slowly.
Rice. 4. Key closure
When the switch is opened (Fig. 5), the resulting self-induction EMF prevents the current from decreasing. Therefore, the current continues to flow for some time. For current to exist, a closed circuit is needed. There is such a circuit in the circuit; it contains both light bulbs. Therefore, when the circuit is opened, the light bulbs should glow the same for some time, and the observed delay may be caused by other reasons.
Rice. 5. Key opening
Let us consider the processes occurring in this circuit when the key is closed and opened.
1. Key closure.
There is a current-carrying coil in the circuit. Let the current in this turn flow counterclockwise. Then the magnetic field will be directed upward (Fig. 6).
Thus, the coil ends up in the space of its own magnetic field. As the current increases, the coil will find itself in the space of a changing magnetic field of its own current. If the current increases, then the magnetic flux created by this current also increases. As is known, with an increase in the magnetic flux penetrating the plane of the circuit, an electromotive force of induction arises in this circuit and, as a consequence, an induction current. According to Lenz's rule, this current will be directed in such a way that its magnetic field prevents a change in the magnetic flux penetrating the plane of the circuit.
That is, for the one considered in Fig. 6 turns, the induction current should be directed clockwise (Fig. 7), thereby preventing the increase in the turn’s own current. Consequently, when the key is closed, the current in the circuit does not increase instantly due to the fact that a braking induction current appears in this circuit, directed in the opposite direction.
2. Opening the key
When the switch is opened, the current in the circuit decreases, which leads to a decrease in the magnetic flux through the plane of the coil. A decrease in magnetic flux leads to the appearance of induced emf and induced current. In this case, the induced current is directed in the same direction as the coil’s own current. This leads to a slower decrease in the intrinsic current.
Conclusion: when the current in a conductor changes, electromagnetic induction occurs in the same conductor, which generates an induced current directed in such a way as to prevent any change in its own current in the conductor (Fig. 8). This is the essence of the phenomenon of self-induction. Self-induction is a special case of electromagnetic induction.
Rice. 8. The moment of switching on and off the circuit
Formula for finding the magnetic induction of a straight conductor with current:
where is magnetic induction; - magnetic constant; - current strength; - distance from the conductor to the point.
The flux of magnetic induction through the area is equal to:
where is the surface area that is penetrated by the magnetic flux.
Thus, the flux of magnetic induction is proportional to the magnitude of the current in the conductor.
For a coil in which is the number of turns and is the length, the magnetic field induction is determined by the following relationship:
Magnetic flux created by a coil with the number of turns N, is equal to:
Substituting the formula for magnetic field induction into this expression, we obtain:
The ratio of the number of turns to the length of the coil is denoted by the number:
We obtain the final expression for the magnetic flux:
From the resulting relationship it is clear that the flux value depends on the current value and on the geometry of the coil (radius, length, number of turns). A value equal to is called inductance:
The unit of inductance is henry:
Therefore, the flux of magnetic induction caused by the current in the coil is equal to:
Taking into account the formula for induced emf, we find that self-induction emf is equal to the product of the rate of change of current and inductance, taken with the “-” sign:
Self-induction- this is the phenomenon of the occurrence of electromagnetic induction in a conductor when the strength of the current flowing through this conductor changes.
Electromotive force of self-induction is directly proportional to the rate of change of current flowing through the conductor, taken with a minus sign. The proportionality factor is called inductance, which depends on the geometric parameters of the conductor.
A conductor has an inductance equal to 1 H if, at a rate of change of current in the conductor equal to 1 A per second, a self-inductive electromotive force equal to 1 V arises in this conductor.
People encounter the phenomenon of self-induction every day. Every time we turn on or off the light, we thereby close or open the circuit, thereby exciting induction currents. Sometimes these currents can reach such high values that a spark jumps inside the switch, which we can see.
Bibliography
- Myakishev G.Ya. Physics: Textbook. for 11th grade general education institutions. - M.: Education, 2010.
- Kasyanov V.A. Physics. 11th grade: Educational. for general education institutions. - M.: Bustard, 2005.
- Gendenstein L.E., Dick Yu.I., Physics 11. - M.: Mnemosyne.
- Internet portal Myshared.ru ().
- Internet portal Physics.ru ().
- Internet portal Festival.1september.ru ().
Homework
- Questions at the end of paragraph 15 (p. 45) - Myakishev G.Ya. Physics 11 (see list of recommended readings)
- The inductance of which conductor is 1 Henry?