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What determines the value of self-induction emf. What is the self-induced emf?

Electricity, passing along the circuit, creates a magnetic field around it. The magnetic flux Φ through the circuit of this conductor (it is called own magnetic flux) is proportional to the induction module B magnetic field inside the circuit \(\left(\Phi \sim B \right)\), and the magnetic field induction in turn is proportional to the current strength in the circuit \(\left(B\sim I \right)\).

Thus, the own magnetic flux is directly proportional to the current strength in the circuit \(\left(\Phi \sim I \right)\). This relationship can be represented mathematically as follows:

\(\Phi = L \cdot I,\)

Where L- proportionality coefficient, which is called circuit inductance.

  • Loop inductance- scalar physical quantity, numerically equal to the ratio of the own magnetic flux penetrating the circuit to the current strength in it:
\(~L = \dfrac(\Phi)(I).\)

The SI unit of inductance is the henry (H):

1 H = 1 Wb/(1 A).

  • The inductance of the circuit is 1 Hn, if at power direct current 1 A magnetic flux through the circuit is 1 Wb.

The inductance of the circuit depends on the size and shape of the circuit, on the magnetic properties of the environment in which the circuit is located, but does not depend on the current strength in the conductor. Thus, the inductance of the solenoid can be calculated using the formula

\(~L = \mu \cdot \mu_0 \cdot N^2 \cdot \dfrac(S)(l),\)

Where μ is the magnetic permeability of the core, μ 0 is the magnetic constant, N- number of solenoid turns, S- coil area, l- solenoid length.

With the shape and dimensions of a fixed circuit remaining unchanged, the intrinsic magnetic flux through this circuit can change only when the current strength in it changes, i.e.

\(\Delta \Phi =L \cdot \Delta I.\) (1)

Self-induction phenomenon

If a direct current passes through a circuit, then there is a constant magnetic field around the circuit, and the intrinsic magnetic flux passing through the circuit does not change over time.

If the current passing in the circuit changes over time, then the correspondingly changing own magnetic flux, and, according to the law of electromagnetic induction, creates an EMF in the circuit.

  • The occurrence of induced emf in a circuit, which is caused by a change in current strength in this circuit, is called self-induction phenomenon. Self-induction was discovered by the American physicist J. Henry in 1832.

The emf that appears in this case is the self-induction emf E si. The self-induction emf creates a self-induction current in the circuit I si.

The direction of the self-induction current is determined by Lenz's rule: the self-induction current is always directed so that it counteracts the change in the main current. If the main current increases, then the self-induction current is directed against the direction of the main current; if it decreases, then the directions of the main current and the self-induction current coincide.

Using the law of electromagnetic induction for an inductive circuit L and equation (1), we obtain the expression for the self-induction emf:

\(E_(si) =-\dfrac(\Delta \Phi )(\Delta t)=-L\cdot \dfrac(\Delta I)(\Delta t).\)

  • The self-induction emf is directly proportional to the rate of change of current in the circuit, taken with the opposite sign. This formula can only be used with a uniform change in current strength. With increasing current (Δ I> 0), negative EMF (E si< 0), т.е. индукционный ток направлен в противоположную сторону тока источника. При уменьшении тока (ΔI < 0), ЭДС положительная (E si >0), i.e. the induced current is directed in the same direction as the source current.

From the resulting formula it follows that

\(L=-E_(si) \cdot \dfrac(\Delta t)(\Delta I).\)

  • Inductance is a physical quantity numerically equal to the self-inductive emf that occurs in the circuit when the current changes by 1 A in 1 s.

The phenomenon of self-induction can be observed in simple experiments. Figure 1 shows a diagram of parallel connection of two identical lamps. One of them is connected to the source through a resistor R, and the other in series with the coil L. When the key is closed, the first lamp flashes almost immediately, and the second with a noticeable delay. This is explained by the fact that in the section of the circuit with the lamp 1 there is no inductance, so there will be no self-induction current, and the current in this lamp almost instantly reaches its maximum value. In the area with the lamp 2 when the current in the circuit increases (from zero to maximum), a self-induction current appears I si, which prevents the rapid increase in current in the lamp. Figure 2 shows an approximate graph of current changes in the lamp 2 when the circuit is closed.

When the key is opened, the current in the lamp 2 will also fade slowly (Fig. 3, a). If the inductance of the coil is large enough, then immediately after opening the switch there may even be a slight increase in current (lamp 2 flares up more strongly), and only then the current begins to decrease (Fig. 3, b).

Rice. 3

The phenomenon of self-induction creates a spark at the point where the circuit opens. If the circuit contains powerful electromagnets, then the spark can turn into an arc and damage the switch. To open such circuits, power plants use special switches.

Magnetic field energy

Magnetic field energy of an inductor circuit L with current strength I

\(~W_m = \dfrac(L \cdot I^2)(2).\)

Since \(~\Phi = L \cdot I\), the energy of the magnetic field of the current (coil) can be calculated knowing any two of the three values ​​( Φ, L, I):

\(~W_m = \dfrac(L \cdot I^2)(2) = \dfrac(\Phi \cdot I)(2)=\dfrac(\Phi^2)(2L).\)

The magnetic field energy contained in a unit volume of space occupied by the field is called bulk density energy magnetic field:

\(\omega_m = \dfrac(W_m)(V).\)

*Derivation of the formula

1 output.

Let's connect a conducting circuit with inductance to a current source L. Let the current increase uniformly from zero to a certain value over a short period of time Δt I (Δ I = I). The self-induction emf will be equal to

\(E_(si) =-L \cdot \dfrac(\Delta I)(\Delta t) = -L \cdot \dfrac(I)(\Delta t).\)

Over a given period of time Δ t charge is transferred through the circuit

\(\Delta q = \left\langle I \right \rangle \cdot \Delta t,\)

where \(\left \langle I \right \rangle = \dfrac(I)(2)\) is the average current value over time Δ t with its uniform increase from zero to I.

Current strength in a circuit with inductance L reaches its value not instantly, but over a certain finite period of time Δ t. In this case, a self-inductive emf E si arises in the circuit, preventing the increase in current strength. Consequently, when the current source is closed, it does work against the self-inductive emf, i.e.

\(A = -E_(si) \cdot \Delta q.\)

The work expended by the source to create current in the circuit (without taking into account thermal losses) determines the magnetic field energy stored by the current-carrying circuit. That's why

\(W_m = A = L \cdot \dfrac(I)(\Delta t) \cdot \dfrac(I)(2) \cdot \Delta t = \dfrac(L \cdot I^2)(2).\ )

2 output.

If the magnetic field is created by the current passing in the solenoid, then the inductance and modulus of the magnetic field of the coil are equal

\(~L = \mu \cdot \mu_0 \cdot \dfrac (N^2)(l) \cdot S, \,\,\, ~B = \dfrac (\mu \cdot \mu_0 \cdot N \cdot I)(l)\)

\(I = \dfrac (B \cdot l)(\mu \cdot \mu_0 \cdot N).\)

Substituting the resulting expressions into the formula for the magnetic field energy, we obtain

\(~W_m = \dfrac (1)(2) \cdot \mu \cdot \mu_0 \cdot \dfrac (N^2)(l) \cdot S \cdot \dfrac (B^2 \cdot l^2) ((\mu \cdot \mu_0)^2 \cdot N^2) = \dfrac (1)(2) \cdot \dfrac (B^2)(\mu \cdot \mu_0) \cdot S \cdot l. \)

Since \(~S \cdot l = V\) is the volume of the coil, the magnetic field energy density is equal to

\(\omega_m = \dfrac (B^2)(2\mu \cdot \mu_0),\)

Where IN- magnetic field induction module, μ - magnetic permeability of the medium, μ 0 - magnetic constant.

Literature

  1. Aksenovich L. A. Physics in high school: Theory. Tasks. Tests: Textbook. allowance for institutions providing general education. environment, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsiya i vyhavanne, 2004. - P. 351-355, 432-434.
  2. Zhilko V.V. Physics: textbook. allowance for 11th grade. general education institutions with Russian language 12-year studies (basic and elevated levels) / V.V. Zhilko, L.G. Markovich. - Mn.: Nar. Asveta, 2008. - pp. 183-188.
  3. Myakishev, G.Ya. Physics: Electrodynamics. 10-11 grades : textbook for in-depth study of physics / G.Ya. Myakishev, A.3. Sinyakov, V.A. Slobodskov. - M.: Bustard, 2005. - P. 417-424.

When the switch is closed in the circuit shown in Figure 1, an electric current will arise, the direction of which is shown by single arrows. With the appearance of current, a magnetic field arises, the induction lines of which cross the conductor and induce an electromotive force (EMF) in it. As stated in the article “The phenomenon of electromagnetic induction”, this EMF is called self-induction EMF. Since any induced emf, according to Lenz’s rule, is directed against the cause that caused it, and this cause will be the emf of the battery of elements, the self-induction emf of the coil will be directed against the emf of the battery. The direction of self-induction EMF in Figure 1 is shown by double arrows.

Thus, the current is not established in the circuit immediately. Only when the magnetic flux is established, the intersection of the conductor with magnetic lines will stop and the self-induction emf will disappear. Then a constant current will flow in the circuit.

Figure 2 shows a graphical representation of direct current. The horizontal axis represents time, along vertical axis- current. It can be seen from the figure that if at the first moment of time the current is 6 A, then at the third, seventh and so on moments of time it will also be equal to 6 A.

Figure 3 shows how the current is established in the circuit after switching on. The self-induction emf, directed at the moment of switching on against the emf of the battery of elements, weakens the current in the circuit, and therefore at the moment of switching on the current is zero. Then, at the first moment of time, the current is 2 A, at the second moment of time - 4 A, at the third - 5 A, and only after some time a current of 6 A is established in the circuit.

Figure 3. Graph of current increase in the circuit taking into account the self-inductive emf Figure 4. The self-induction EMF at the moment of opening the circuit is directed in the same direction as the EMF of the voltage source

When the circuit is opened (Figure 4), the disappearing current, the direction of which is shown by a single arrow, will reduce its magnetic field. This field, decreasing from a certain value to zero, will again cross the conductor and induce a self-induction emf in it.

When turning off electrical circuit with inductance, the self-inductive emf will be directed in the same direction as the emf of the voltage source. The direction of the self-induction EMF is shown in Figure 4 by a double arrow. As a result of the action of self-induction emf, the current in the circuit does not disappear immediately.

Thus, the self-induced emf is always directed against the cause that caused it. Noting this property, they say that the self-induction EMF is reactive in nature.

Graphically, the change in current in our circuit, taking into account the self-inductive emf when it is closed and when it is subsequently opened at the eighth moment in time, is shown in Figure 5.

Figure 5. Graph of the rise and fall of the current in the circuit, taking into account the self-induction emf Figure 6. Induction currents when the circuit is opened

When opening circuits containing a large number of turns and massive steel cores or, as they say, having high inductance, the self-inductive emf can be many times greater than the emf of the voltage source. Then, at the moment of opening, the air gap between the knife and the fixed clamp of the switch will be broken and the resulting electric arc will melt the copper parts of the switch, and if there is no casing on the switch, it can burn a person’s hands (Figure 6).

In the circuit itself, the self-induction EMF can break through the insulation of the turns of coils, electromagnets, and so on. To avoid this, some switching devices provide protection against self-induction EMF in the form of a special contact that short-circuits the electromagnet winding when switched off.

It should be taken into account that the self-induction EMF manifests itself not only at the moments when the circuit is turned on and off, but also during any changes in current.

The magnitude of the self-induction emf depends on the rate of change of current in the circuit. So, for example, if for the same circuit in one case within 1 second the current in the circuit changed from 50 to 40 A (that is, by 10 A), and in another case from 50 to 20 A (that is, by 30 A ), then in the second case a threefold greater self-induction emf will be induced in the circuit.

The magnitude of the self-inductive emf depends on the inductance of the circuit itself. Circuits with high inductance are the windings of generators, electric motors, transformers and induction coils with steel cores. Straight conductors have lower inductance. Short straight conductors, incandescent lamps and electric heating devices (stoves, stoves) have practically no inductance and the appearance of self-inductive emf in them is almost not observed.

The magnetic flux penetrating the circuit and inducing the self-induction emf in it is proportional to the current flowing through the circuit:

F = L × I ,

Where L- proportionality coefficient. It's called inductance. Let us determine the dimension of inductance:

Ohm × sec is otherwise called henry (Hn).

1 henry = 10 3 ; millihenry (mH) = 10 6 microhenry (µH).

Inductance, except Henry, is measured in centimeters:

1 henry = 10 9 cm.

For example, 1 km of telegraph line has an inductance of 0.002 H. The inductance of the windings of large electromagnets reaches several hundred henries.

If the loop current changes by Δ i, then the magnetic flux will change by the value Δ Ф:

Δ Ф = L × Δ i .

The magnitude of the self-induction EMF that appears in the circuit will be equal to (formula of the self-induction EMF):

If the current changes uniformly over time, the expression will be constant and can be replaced by the expression. Then the absolute value of the self-induction emf arising in the circuit can be found as follows:

Based on the last formula, we can define the unit of inductance - henry:

A conductor has an inductance of 1 H if, with a uniform change in current by 1 A per 1 second, a self-inductive emf of 1 V is induced in it.

As we saw above, self-induction emf occurs in a direct current circuit only at the moments of its switching on, switching off, and whenever it changes. If the magnitude of the current in the circuit is unchanged, then the magnetic flux of the conductor is constant and the self-induction emf cannot arise (since. At moments of change in the current in the circuit, the self-induction emf interferes with changes in the current, that is, it provides a kind of resistance to it.

This phenomenon is called self-induction. (The concept is related to the concept of mutual induction, being, as it were, a special case of it).

The direction of the self-induction EMF always turns out to be such that when the current in the circuit increases, the self-induction EMF prevents this increase (directed against the current), and when the current decreases, it decreases (co-directed with the current). This property of self-induction emf is similar to inertial force.

The magnitude of the self-induction EMF is proportional to the rate of change of current:

.

The proportionality factor is called self-induction coefficient or inductance circuit (coil).

Self-induction and sinusoidal current

In the case of a sinusoidal dependence of the current flowing through the coil on time, the self-inductive emf in the coil lags behind the current in phase by (that is, 90°), and the amplitude of this emf is proportional to the amplitude of the current, frequency and inductance (). After all, the rate of change of a function is its first derivative, a.

To calculate more or less complex circuits containing inductive elements, that is, turns, coils, etc. devices in which self-induction is observed (especially completely linear ones, that is, not containing nonlinear elements), in the case of sinusoidal currents and voltages, the method of complex impedances is used or, in simpler cases, a less powerful, but more visual option is the vector diagram method.

Note that everything described applies not only directly to sinusoidal currents and voltages, but also practically arbitrary, since the latter can almost always be expanded into a series or Fourier integral and thus reduced to sinusoidal.

In more or less direct connection with this, we can mention the use of the phenomenon of self-induction (and, accordingly, inductors) in a variety of oscillating circuits, filters, delay lines and other various electronics and electrical circuits.

Self-inductance and current surge

Due to the phenomenon of self-induction in an electrical circuit with an EMF source, when the circuit is closed, the current is not established instantly, but after some time. Similar processes occur when the circuit opens, and (with a sharp opening) the value of the self-induction EMF at this moment can significantly exceed the source EMF.

Most often in everyday life this is used in car ignition coils. Typical ignition voltage with a 12V battery voltage is 7-25 kV. However, the excess of the EMF in the output circuit over the EMF of the battery here is due not only to a sharp interruption of the current, but also to the transformation ratio, since most often it is not used simple coil inductance, and the coil is a transformer, secondary winding which usually has many times large quantity turns (that is, in most cases the circuit is somewhat more complex than the one whose operation would be fully explained through self-induction; however, the physics of its operation in this version partly coincides with the physics of the operation of a circuit with a simple coil).

This phenomenon is also used for ignition. fluorescent lamps in a standard traditional circuit (here we are talking specifically about a circuit with a simple inductor - a choke).

In addition, it must always be taken into account when opening contacts, if the current flows through the load with noticeable inductance: the resulting jump in EMF can lead to breakdown of the intercontact gap and/or other undesirable effects, to suppress which in this case, as a rule, it is necessary to take a variety of special measures.

Notes

Links

  • About self-induction and mutual induction from the “School for Electricians”

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See what “Self-induction” is in other dictionaries:

    Self-induction... Spelling dictionary-reference book

    The appearance of induced emf in a conductive circuit when the current strength changes in it; special cases of electromagnetic induction. When the current in the circuit changes, the magnetic flux changes. induction through the surface limited by this contour, resulting in ... Physical encyclopedia

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    SELF-INDUCTION, self-induction, female. (physical). 1. units only The phenomenon that when the current changes in a conductor, an electromotive force appears in it, preventing this change. Self-induction coil. 2. A device with... ... Dictionary Ushakova

    - (Self induction) 1. A device with inductive reactance. 2. The phenomenon that when an electric current changes in magnitude and direction in a conductor, an electromotive force appears in it, preventing this... ... Marine Dictionary

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    Self-induction- electromagnetic induction caused by a change in the magnetic flux interlocking with the circuit, caused by the electric current in this circuit... Source: ELECTRICAL ENGINEERING. TERMS AND DEFINITIONS OF BASIC CONCEPTS. GOST R 52002 2003 (approved... ... Official terminology

    Noun, number of synonyms: 1 excitation of electromotive force (1) Dictionary of synonyms ASIS. V.N. Trishin. 2013… Synonym dictionary

    self-induction- Electromagnetic induction caused by a change in the magnetic flux interlocking with the circuit, caused by the electric current in this circuit. [GOST R 52002 2003] EN self induction electromagnetic induction in a tube of current due to variations… … Technical Translator's Guide

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Books

  • Set of tables. Physics. Electrodynamics (10 tables), . Educational album of 10 sheets. Electric current, current strength. Resistance. Ohm's law for a section of a circuit. Dependence of conductor resistance on temperature. Connection of wires. EMF. Ohm's law…

9.4. The phenomenon of electromagnetic induction

9.4.3. Average value electromotive force self-induction

When a flow linked to a closed conducting contour changes through the area limited by this contour, a vortex appears in it electric field and an induction current flows - the phenomenon of electromagnetic self-induction.

Module average self-induction emf for a certain period of time is calculated using the formula

〈 | ℰ i s | 〉 = | Δ Ф s | Δt,

where ΔФ s is the change in the magnetic flux coupled to the circuit during the time Δt.

If the current strength in the circuit changes over time I = I (t), then

∆Ф s = L ∆I,

where L is the inductance of the circuit; ΔI - change in current strength in the circuit over time Δt;

〈 | ℰ i s | 〉 = L | ΔI | Δt,

where ΔI /Δt is the rate of change of current in the circuit.

If loop inductance changes over time L = L (t), then

  • the change in flow coupled to the contour is determined by the formula

∆Ф s = ∆LI,

where ΔL is the change in circuit inductance over time Δt; I - current strength in the circuit;

  • the module of the average self-induction emf for a certain period of time is calculated by the formula

〈 | ℰ i s | 〉 = I | Δ L | Δt.

Example 16. In a closed conducting circuit with an inductance of 20 mH, a current of 1.4 A flows. Find the average value of the self-induction emf that occurs in the circuit when the current in it is uniformly reduced by 20% in 80 ms.

Solution . The appearance of self-induction emf in a circuit is caused by a change in the flux coupled to the circuit when the current strength in it changes.

The flow associated with the circuit is determined by the formulas:

  • at current strength I 1

Ф s 1 = LI 1,

where L is the circuit inductance, L = 20 mH; I 1 - initial current in the circuit, I 1 = 1.4 A;

  • at current strength I 2

Ф s 2 = LI 2,

where I 2 is the final current strength in the circuit.

The change in the flow coupled to the circuit is determined by the difference:

Δ Ф s = Ф s 2 − Ф s 1 = L I 2 − L I 1 = L (I 2 − I 1) ,

where I 2 = 0.8I 1.

The average value of the self-induction emf that occurs in the circuit when the current strength changes in it:

〈 ℰ s i 〉 = | Δ Ф s Δ t | = | L (I 2 − I 1) Δ t | = | − 0.2 L I 1 Δ t | = 0.2 L I 1 Δ t,

where ∆t is the time interval during which the current decreases, ∆t = 80 ms.

The calculation gives the value:

〈 ℰ s i 〉 = 0.2 ⋅ 20 ⋅ 10 − 3 ⋅ 1.4 80 ⋅ 10 − 3 = 70 ⋅ 10 − 3 s = 70 mV.

When the current in the circuit changes, a self-inductive emf appears in it, the average value of which is 70 mV.