The phenomenon of electromagnetic induction
1. Faraday's experiments. Basic law of electromagnetic
induction.
1. Faraday's experiments. Basic law of electromagnetic induction.
In 1831, M. Faraday established through numerous experiments that in a closed conducting circuit, when the magnetic flux changes through the surface limited by this circuit, an electric current arises. 
Electromagnetic induction (EMI)– the phenomenon of the occurrence of electric current in a closed conducting circuit when the magnetic flux changes through the surface limited by this circuit.
The appearance of electric current (called induced current) in a closed conducting circuit when the magnetic field penetrating the circuit changes, indicates the action of external forces of non-electrostatic origin in the circuit or the occurrence induced emf.
The magnitude of the induction current is determined by the rate of change of the magnetic flux F, that is, the value, and does not depend on the way the magnetic flux changes F. When the sign changes, the direction of the induction current also changes. 
The general rule by which the direction of the induction current can be determined and which is a consequence of the law of conservation and transformation of energy was formulated by E.Kh. Lenz.
Lenz's rule: the induced current in a closed conducting loop always has such a direction that the magnetic field it creates prevents the change in the external magnetic flux that caused this induced current. Or in short: the induced current is always directed in such a way as to counteract the cause that caused it.
Induction current, like any electric current, can flow in a circuit only if there is an electromotive force in it. Faraday established that the magnitude of the induced emf is directly proportional to the rate of change of the magnetic flux.
Faraday's Basic Law of EMR: The induced emf in a conducting circuit is directly proportional to the rate of change of magnetic flux through the surface bounded by the circuit:
The minus sign serves as a mathematical expression of Lenz's rule, that is, it indicates that the electromotive force counteracts the ongoing change in magnetic flux.
If the circuit in which the EMF is induced consists of N identical turns, then the EMF of such a circuit will be equal to the sum of the induced EMF in each of the turns separately:
Mechanisms of occurrence of induced emf:
– the action of the Lorentz force on charges in a moving conductor;
– the action of a vortex electric field on charges in a conductor.
Induction emf arising in a linear conductor moving in a magnetic field:
Induction currents arise not only in linear conductors, but also in massive solid conductors. These currents are closed inside the conductor and are therefore called vortex currents or Foucault's currents.
Eddy currents, due to the low resistance of a solid conductor, can reach very high strengths. Their thermal effect is used in induction furnaces for heating when hardening parts. Foucault currents obey Lenz's rule, so good conductors moving in a strong magnetic field experience strong braking due to the interaction of eddy currents with the magnetic field. This is used to calm the moving parts of galvanometers and other instruments. In many cases, Foucault currents are unwanted, and special measures have to be taken to combat them (for example, transformer cores are made of thin plates).
2. Self-induction. Mutual induction.
The phenomenon of self-induction is a special case of electromagnetic induction. This phenomenon consists in the occurrence of induced emf in a conductor due to a change in magnetic flux caused by an electric current in the same conductor.
Self-induction– the phenomenon of the occurrence of induced emf in a conductor when the current strength in it changes.
An electric current in a circuit creates a magnetic field around itself, induction IN which, according to the Biot-Savart-Laplace law, with constant magnetic permeability, constant shape and orientation of the contour in space, is proportional to the current strength I:
B~I.
Magnetic flux F through the circuit is proportional by definition of induction IN: F ~ V.
Therefore, the magnetic flux through the loop is proportional to the current in the loop:
Proportionality factor L called circuit inductance. Inductance depends on the size and shape of the conductor, the magnetic permeability of the environment in which it is located. In the SI system:
Self-induced emf, arising in a circuit with inductance L, according to the EMR law is equal to:
The self-inductive emf is directly proportional to the inductance and the rate of change of current in the circuit. The minus sign expresses Lenz's rule: when the current increases, the self-inductive emf is directed towards it, and when it decreases, it maintains the current in the same direction.
The phenomenon of self-induction manifests itself with any change in current strength and therefore plays a very important role in alternating current circuits and in the processes of electromagnetic oscillations.
The phenomenon of self-induction can be observed by assembling the following electrical circuit.
When the current source is turned on, lamp L 1 flashes instantly, and lamp L 2 flashes after a certain period of time.
When the current source is turned off, both lamps L 1 and L 2 go out after a certain period of time.
Self-induction currents arising in a direct current circuit at the moments of closing and opening the circuit are called fault currents And opening.
When the circuit is closed, the current changes according to the law:
and when the circuit opens - according to the law:
Where R– circuit resistance, – steady-state current.
When the source is turned off, the current in the circuit arises under the influence of self-induction emf. The source of energy released in the electrical circuit is the magnetic field of the coil. The energy of the magnetic field is equal to the work expended by the current to create this field:
Hence, magnetic field energy will be equal to:
The phenomenon of mutual induction is another special case of electromagnetic induction.
Mutual induction– the phenomenon of the occurrence of induced emf in a circuit located in the magnetic field of another circuit with alternating current.
When current flows in circuit 1 I 1 In circuit 2, an induced emf occurs:
Similarly, when current flows in circuit 2 I 2 In circuit 1, an induced emf occurs:
Proportionality coefficients, Gn are called mutual inductance of the circuits. They depend on the size, shape, location of the circuits and on the magnetic permeability of the environment in which the circuits are located.
The principle of operation of a transformer is based on the phenomenon of mutual induction.
Transformer- a device used to increase or decrease alternating current voltage (P.N. Yablochkov, 1878).
Primary winding Secondary winding
N 1 ← number of turns → N 2
The attitude is called transformation ratio.
At k 1 transformer is increasing, and when k – downward.
3. Operating principle of the current generator.
Current generator– a device designed to convert mechanical energy into electrical energy.
The principle of operation of a current generator based on the EMR phenomenon can be considered using the example of a flat frame rotating in a uniform magnetic field between the poles of a magnet.
Magnetic flux through an area S framework:
, ω – angular speed of rotation of the frame.
Induction EMF in the frame:
– amplitude of EMF oscillations.
To enhance the effect, frames with a large number of turns are used N. Then:
The induced emf changes according to the sine law.
Lesson results
Control questions
1. What is the phenomenon of electromagnetic induction? Analyze Faraday's experiments.
2. What causes the occurrence of induced emf in a closed conducting circuit?
3. Why is it better to use a closed conductor in the form of a coil to detect induced current, rather than in the form of a single turn of wire?
4. Formulate Lenz’s rule, illustrating it with examples.
5. What are eddy currents (Foucault currents)? Are they harmful or beneficial?
6. Why are transformer cores not made solid?
7. What are the phenomena of self-induction and mutual induction?
8. What physical quantity is expressed in henry? Define Henry.
9. What is a current generator?
10. Derive an expression for the induced emf in a flat frame rotating uniformly in a uniform magnetic field. How can it be increased?
The phenomenon of electromagnetic induction was discovered by Faraday in 1831. Faraday's experiments showed that in any closed conducting circuit, when the number changes
lines of magnetic induction passing through it, an electric current arises. This current was named induced current. For example, at the moment the magnet is inserted and at the moment it is pulled out of the coil, a deflection of the galvanometer needle is observed. The deflections of the arrow when moving in and out are opposite. The faster the magnet moves, the greater the deviations. If you move the magnet into the coil with the other pole, the needle deflections will be opposite to the original ones.
In another experiment, one of the coils K1 is inside another coil K2. When the current through coil K1 is turned on or off, or when it changes, or when the coils move relative to each other, a deflection of the galvanometer needle is observed if current flows through K1.
The total number of lines of magnetic induction through the area of the circuit is magnetic flux. Thus, The cause of the induced current is a change in the magnetic flux through the circuit . If the circuit is located in a uniform magnetic field, the induction of which is equal to B, then the magnetic flux through the circuit, the area of which is S
:Φ = B·Scosα (3.10)
Where α angle between vector IN and normal n to the contour surface.
Magnetic flux is a scalar quantity. If the vector lines IN exit the platform, the magnetic flux is considered positive, if they enter it, the magnetic flux is considered negative. The SI unit of magnetic flux is the weber (Wb).
One weber is a magnetic flux created by a uniform magnetic field of induction 1 T through an area of 1 m² perpendicular to the induction lines. 1Wb = 1T m².
The occurrence of an induced current means that when the magnetic flux Φ changes in the circuit, an induced emf occurs. It is determined by the rate of change of the magnetic flux, i.e.
e = – ΔΦ / Δt (3.11)
Formula (3.11) expresses Faraday's law. The minus sign is a mathematical expression of Lenz's rule, which states that induced current is always directed so as to counteract the cause that causes it .
In other words:
The induced current creates a magnetic flux that prevents the change in magnetic flux that causes induced emf .
LAW OF ELECTROMAGNETIC INDUCTION. LENTZ'S RULE
In 1831, the English physicist M. Faraday discovered the phenomenon of electromagnetic induction in his experiments. Then the Russian scientist E.Kh. studied this phenomenon. Lenz and B. S. Jacobi.
Currently, many devices are based on the phenomenon of electromagnetic induction, for example in a motor or electric current generator, in transformers, radio receivers, and many other devices.
Electromagnetic induction is the phenomenon of the occurrence of current in a closed conductor when a magnetic flux passes through it.
That is, thanks to this phenomenon we can convert mechanical energy into electrical energy. Before the discovery of this phenomenon, people did not know about methods of producing electric current other than electroplating.
When a conductor is exposed to a magnetic field, an emf arises in it, which can be quantitatively expressed through the law of electromagnetic induction.
Law of Electromagnetic Induction
The electromotive force induced in a conducting circuit is equal to the rate of change of the magnetic flux coupling to that circuit.
In a coil that has several turns, the total emf depends on the number of turns n:
The EMF excited in the circuit creates a current. The simplest example of the appearance of current in a conductor is a coil through which a permanent magnet passes. The direction of the induced current can be determined using Lenz's rule.
Lenz's rule
The current induced by a change in the magnetic field passing through the circuit prevents this change with its magnetic field.
In the case when we introduce a magnet into the coil, the magnetic flux in the circuit increases, which means that the magnetic field created by the induced current, according to Lenz’s rule, is directed against the increase in the magnet’s field. To determine the direction of the current, you need to look at the magnet from the north pole. From this position we will screw the gimlet in the direction of the magnetic field of the current, that is, towards the north pole. The current will move in the direction of rotation of the gimlet, that is, clockwise.
In the case when we remove the magnet from the coil, the magnetic flux in the circuit decreases, which means the magnetic field created by the induced current is directed against the decrease in the magnet's field. To determine the direction of the current, you need to unscrew the gimlet; the direction of rotation of the gimlet will indicate the direction of the current in the conductor - counterclockwise.
An electric generator is a device in which non-electrical types of energy (mechanical, chemical, thermal) are converted into electrical energy.
Classification of electromechanical generators
By type of prime mover:
Turbogenerator - an electrical generator driven by a steam turbine or gas turbine engine;
Hydrogenerator - an electric generator driven by a hydraulic turbine;
Diesel generator - an electric generator driven by a diesel engine;
Wind generator - an electric generator that converts the kinetic energy of the wind into electricity;
According to the type of output electric current
Three-phase generator with star windings
With triangle windings included
According to the method of excitation
Excited by permanent magnets
With external excitation
Self-excited
With sequential excitation
With parallel excitation
With mixed excitement
According to the principle of operation, generators can be synchronous or asynchronous.
Asynchronous generators are structurally simple and inexpensive to manufacture, and are more resistant to short circuit currents and overloads. An asynchronous electric generator is ideal for powering active loads: incandescent lamps, electric heaters, electronics, electric burners, etc. But even short-term overload is unacceptable for them, therefore, when connecting electric motors, non-electronic welding machines, power tools and other inductive loads, there is a reserve of power should be at least three times, and preferably four times.
A synchronous generator is perfect for inductive consumers with high starting currents. They are capable of withstanding a fivefold current overload for one second.
Operating principle of the current generator
The generator operates on the basis of Faraday's law of electromagnetic induction - electromotive force (EMF) is induced in a rectangular loop (wire frame) rotating in a uniform magnetic field.
EMF also occurs in a stationary rectangular frame if a magnet is rotated in it.
The simplest generator is a rectangular frame placed between 2 magnets with different poles. In order to remove the voltage from the rotating frame, slip rings are used.
A car generator consists of a housing and two covers with holes for ventilation. The rotor rotates in 2 bearings and is driven by a pulley. At its core, the rotor is an electromagnet consisting of one winding. Current is supplied to it using two copper rings and graphite brushes, which are connected to an electronic relay controller. He is responsible for ensuring that the voltage supplied by the generator is always within the permissible limits of 12 Volts with permissible deviations and does not depend on the pulley rotation speed. The relay regulator can be either built into the generator housing or located outside it.
The stator consists of three copper windings interconnected in a triangle. A rectifier bridge of 6 semiconductor diodes is connected to their connection points, which convert the voltage from AC to DC.
A gasoline electric generator consists of an engine and a current generator driving it directly, which can be either synchronous or asynchronous.
The engine is equipped with systems: starting, fuel injection, cooling, lubrication, speed stabilization. Vibration and noise are absorbed by a muffler, vibration dampers and shock absorbers.
Alternating electric current
Electromagnetic vibrations, like mechanical ones, are of two types: free and forced.
Free electromagnetic oscillations, always damped oscillations. Therefore, in practice they are almost never used. While forced vibrations are used everywhere and everywhere. Every day you and I can observe these fluctuations.
All our apartments are lit using alternating current. Alternating current is nothing more than forced electromagnetic oscillations. The current and voltage will change over time according to the harmonic law. Fluctuations, for example, in voltage can be detected by applying voltage from an outlet to an oscilloscope.
A sine wave will appear on the oscilloscope screen. The frequency of alternating current can be calculated. It will be equal to the frequency of electromagnetic oscillations. The standard frequency for industrial alternating current is assumed to be 50 Hz. That is, in 1 second the direction of the current in the socket changes 50 times. US industrial networks use a frequency of 60 Hz.
A change in voltage at the ends of the circuit will cause a change in the current strength in the oscillatory circuit circuit. It should still be understood that the change in the electric field in the entire circuit does not occur instantly.
But since this time is significantly less than the period of voltage oscillation at the ends of the circuit, it is usually believed that the electric field in the circuit immediately changes as the voltage at the ends of the circuit changes.
The alternating voltage in the outlet is created by generators in power plants. The simplest generator can be considered a wire frame that rotates in a uniform magnetic field.
The magnetic flux penetrating the circuit will constantly change and will be proportional to the cosine of the angle between the magnetic induction vector and the normal to the frame. If the frame rotates uniformly, the angle will be proportional to time.
Consequently, the magnetic flux will change according to the harmonic law:
Ф = B*S*cos(ω*t)
The rate of change of magnetic flux, taken with the opposite sign, according to the EMR law, will be equal to the induced emf.
Ei = -Ф’ = Em*sin(ω*t).
If an oscillatory circuit is connected to the frame, the angular speed of rotation of the frame will determine the frequency of voltage oscillations in different sections of the circuit and the current strength. In what follows, we will consider only forced electromagnetic oscillations.
They are described by the following formulas:
u = Um*sin(ω*t),
u = Um*cos(ω*t)
Here Um is the amplitude of voltage fluctuations. Voltage and current change with the same frequency ω. But voltage fluctuations will not always coincide with current fluctuations, so it is better to use a more general formula:
I = Im*sin(ω*t +φ), where Im is the amplitude of current fluctuations, and φ is the phase shift between current and voltage fluctuations.
AC current and voltage parameters
The magnitude of alternating current, like voltage, constantly changes over time. Quantitative indicators for measurements and calculations use their following parameters:
Period T is the time during which one complete cycle of current change occurs in both directions relative to zero or the average value.
Frequency f is the reciprocal of the period, equal to the number of periods in one second. One period per second is one hertz (1 Hz)
f = 1/T
Cyclic frequency ω - angular frequency equal to the number of periods in 2π seconds.
ω = 2πf = 2π/T
Typically used in sinusoidal current and voltage calculations. Then, within the period, one can not consider frequency and time, but make calculations in radians or degrees. T = 2π = 360°
The initial phase ψ is the value of the angle from zero (ωt = 0) to the beginning of the period. Measured in radians or degrees. Shown in the figure for a blue sinusoidal current graph. The initial phase can be a positive or negative value, respectively to the right or left of zero on the graph.
Instantaneous value - the value of voltage or current measured relative to zero at any selected time t.
i = i(t); u = u(t)
The sequence of all instantaneous values in any time interval can be considered as a function of the change in current or voltage over time. For example, a sinusoidal current or voltage can be expressed by the function:
i = Iampsin(ωt); u = Uampsin(ωt)
Taking into account the initial phase:
i = Iampsin(ωt + ψ); u = Uampsin(ωt + ψ)
Here Iamp and Uamp are the amplitude values of current and voltage.
Amplitude value is the maximum absolute instantaneous value for the period.
Iamp = max|i(t)|; Uamp = max|u(t)|
Can be positive or negative depending on its position relative to zero. Often, instead of the amplitude value, the term current (voltage) amplitude is used - the maximum deviation from the zero value.
D/z
Report on the topic (of the student's choice)
Electricity generation and transmission
Transformer. Transmission of electricity over a distance
Energy saving in everyday lifeFirst experiments in transmitting electricity over a distance Transformer efficiency. Design and operationUse of electricityTurbogenerator. Design and operation
Hydrogenerator. Design and operation
Diesel generator. Design and operation
Wind generator. Design and operation
Problems to solve independently
Faraday's law of EM induction.
1. The magnetic flux inside a coil with a number of turns equal to 400 changed from 0.1 Wb to 0.9 Wb in 0.2 s. Determine the emf induced in the coil.
2. Determine the magnetic flux passing through a rectangular area with sides of 20x40 cm, if it is placed in a uniform magnetic field with an induction of 5 Tesla at an angle of 60° to the lines of magnetic induction of the field.
3. How many turns should the coil have so that when the magnetic flux inside it changes from 0.024 to 0.056 Wb in 0.32 s, an average emf is created in it. 10 V?
Induction emf in moving conductors.
1. Determine the induced emf at the ends of the wings of the An-2 aircraft, having a length of 12.4 m, if the speed of the aircraft in horizontal flight is 180 km/h, and the vertical component of the induction vector of the Earth’s magnetic field is 0.5·10-4 T.
2. Find the induced emf on the wings of a Tu-204 aircraft, having a length of 42 m, flying horizontally at a speed of 850 km/h, if the vertical component of the induction vector of the Earth’s magnetic field is 5·10-5 T.
Self-induced emf
1. A magnetic flux of 0.015 Wb appears in a coil when a current of 5.0 A passes through its turns. How many turns does the coil contain if its inductance is 60 mH?
2. How many times will the inductance of a coil without a core change if the number of turns in it is doubled?
3. What is the e.m.f. self-induction will occur in a coil with an inductance of 68 mH if a current of 3.8 A disappears in it in 0.012 s?
4. Determine the inductance of the coil if, when the current in it is weakened by 2.8 A, an average emf appears in the coil in 62 ms. self-induction 14 V.
5. How long does it take in a coil with an inductance of 240 mH to increase the current from zero to 11.4 A, if an average emf occurs? self-induction 30 V?
Electromagnetic field energy
1. A current of 20 A flows through a coil with an inductance of 0.6 H. What is the energy of the magnetic field of the coil? How will this energy change when the current increases by a factor of 2? 3 times?
2. How much current must be passed through the winding of an inductor with an inductance of 0.5 H so that the field energy is equal to 100 J?
3. The energy of the magnetic field of which coil is greater and by how many times, if the first has the characteristics: I1=10A, L1=20 H, the second: I2=20A, L2=10 H?
4. Determine the energy of the magnetic field of the coil in which, at a current of 7.5 A, the magnetic flux is 2.3·10-3 Wb. The number of turns in the coil is 120.
5. Determine the inductance of the coil if, at a current of 6.2 A, its magnetic field has an energy of 0.32 J.
6. The magnetic field of a coil with an inductance of 95 mH has an energy of 0.19 J. What is the current strength in the coil?