Formulas of electromagnetic induction 11. Faraday's law of electromagnetic induction

The law of electromagnetic induction (z. Faraday-Maxwell). Lenz's rules

Summarizing the result of the experiments, Faraday formulated the law of electromagnetic induction. He showed that with any change in the magnetic flux in a closed conducting circuit, an induction current is excited. Therefore, an induction emf occurs in the circuit.

The induction emf is directly proportional to the rate of change of the magnetic flux over time. The mathematical record of this law was designed by Maxwell and therefore it is called the Faraday-Maxwell law (the law of electromagnetic induction).

4.2.2. Lenz's rule

The law of electromagnetic induction does not say about the direction of the inductive current. This question was solved by Lenz in 1833. He established a rule to determine the direction of the induction current.

The induction current has such a direction that the magnetic field created by it prevents a change in the magnetic flux penetrating this circuit, i.e. induction current. It is directed in such a way as to counteract the cause that causes it. For example, let a permanent magnet NS be pushed into a closed circuit (Fig. 250).

Fig.250 Fig.251

The number of lines of force crossing the closed circuit increases, therefore, the magnetic flux increases. In the circuit there is an induction current I i , which creates a magnetic field, the lines of force of which (dotted lines perpendicular to the plane of the contour) are directed against the lines of force of the magnet. When the magnet is extended, the magnetic flux penetrating the circuit decreases (Fig. 251), and the induction current I i creates a field, the lines of force of which are directed towards the line of induction of the magnet (dashed lines in Fig. 251).

Taking into account the Lenz rule, the Faraday-Maxwell law can be written in the form

Formula (568) is used to solve a physical problem.

The time-averaged value of the induction emf is determined by the formula

Find out ways to change the magnetic flux.

First way. B=const and α=const. Area changes S.

Example. Let in a uniform magnetic field B=const a conductor of length l moves perpendicular to the lines of force with a speed (Fig. 252) Then a potential difference arises at the ends of the conductor, equal to the EMF of induction. Let's find her.

The change in magnetic flux is

In the formula (570) α - this is the angle between the normal of the plane, washed by the movement of the conductor, and the induction vector.

After it was established that the magnetic field is created by electric currents, scientists tried to solve the inverse problem - using a magnetic field to create an electric current. This problem was successfully solved in 1831 by M. Faraday, who discovered the phenomenon of electromagnetic induction. The essence of this phenomenon is that in a closed conducting circuit, with any change in the magnetic flux penetrating this circuit, an electrical current arises, which is called induction. A diagram of some of Faraday's experiments is shown in fig. 3.12.

When the position of the permanent magnet was changed relative to the coil closed to the galvanometer, an electric current arose in the latter, and the direction of the current turned out to be different - depending on the direction of movement of the permanent magnet. A similar result was achieved when moving another coil, through which an electric current flowed. Moreover, a current appeared in the large coil even with the position of the smaller coil unchanged, but with a change in the current in it.

Based on such experiments, M. Faraday came to the conclusion that an electric current always appears in the coil when the magnetic flux coupled to this coil changes. The magnitude of the current depends on the rate of change of the magnetic flux. Now we formulate Faraday's discoveries in the form law of electromagnetic induction: with any change in the magnetic flux coupled to a conducting closed circuit, an EMF of induction arises in this circuit, which is defined as

The “-” sign in expression (3.53) means that with an increase in the magnetic flux, the magnetic field created by the induction current is directed against the external magnetic field. If the magnetic flux decreases in magnitude, then the magnetic field of the induction current coincides in direction with the external magnetic field. The Russian scientist H. Lenz thus determined the appearance of the minus sign in the expression (3.53) - the induction current in the circuit always has such a direction that the magnetic field created by it has such a direction that it prevents a change in the magnetic flux that caused the induction current to occur.

Let's give another wording. law of electromagnetic induction: The induction emf in a closed conducting circuit is equal to the rate of change of the magnetic flux penetrating this circuit, taken with the opposite sign.

The German physicist Helmholtz showed that the law of electromagnetic induction can be obtained from the law of conservation of energy. In fact, the energy of the EMF source to move the current-carrying conductor in a magnetic field (see Fig. 3.37) will be spent both on the Joule heating of the conductor with resistance R, and on the work of moving the conductor:

Then it immediately follows from equation (3.54) that

The numerator of expression (3.55) is the algebraic sum of the EMF acting in the circuit. Hence,

What is the physical reason for the occurrence of EMF? The Lorentz force acts on the charges in the conductor AB when the conductor moves along the x axis. Under the action of this force, the positive charges will shift upward, as a result of which the electric field in the conductor will be weakened. In other words, an EMF of induction will appear in the conductor. Therefore, in the case considered by us, the physical cause of the emf is the Lorentz force. However, as we have already noted, an EMF of induction may appear in a stationary closed circuit if the magnetic field penetrating this circuit changes.

In this case, the charges can be considered fixed, and the Lorentz force does not act on fixed charges. To explain the occurrence of EMF in this case, Maxwell suggested that any changing magnetic field generates a changing electric field in the conductor, which is the cause of the induction EMF. The circulation of the voltage vector acting in this circuit will thus be equal to the induction EMF acting in the circuit:

. (3.56)

The phenomenon of electromagnetic induction is used to convert mechanical rotational energy into electrical energy - in electric current generators. The reverse process - the conversion of electrical energy into mechanical energy, based on the torque acting on the frame with current in a magnetic field, is used in electric motors.

Consider the principle of operation of the electric current generator (Fig. 3.13). Let us have a conducting frame rotate between the poles of a magnet (it can be an electromagnet) with a frequency w. Then the angle between the normal to the plane of the frame and the direction of the magnetic field changes according to the law a = wt. In this case, the magnetic flux coupled to the frame will change in accordance with the formula

where S is the area of ​​the contour. In accordance with the law of electromagnetic induction, an EMF will be induced in the frame

with e max = BSw. Thus, if a conducting frame rotates at a constant angular velocity in a magnetic field, then an EMF will be induced in it, changing according to a harmonic law. In real generators, many turns connected in series are rotated, and in electromagnets, to increase the magnetic induction, cores with high magnetic permeability are used. m..

Induction currents can also occur in the thickness of conducting bodies placed in an alternating magnetic field. In this case, these currents are called Foucault currents. These currents cause heating of massive conductors. This phenomenon is used in vacuum induction furnaces, where high currents heat the metal to melt. Since the heating of metals occurs in a vacuum, this makes it possible to obtain highly pure materials.

The magnetic induction vector \(~\vec B\) characterizes the force properties of the magnetic field at a given point in space. Let us introduce one more quantity that depends on the value of the magnetic induction vector not at one point, but at all points of an arbitrarily chosen surface. This value is called magnetic flux and is denoted by the Greek letter Φ (phi).

• magnetic fluxΦ of a uniform field through a flat surface is a scalar physical quantity numerically equal to the product of the modulus of induction B magnetic field, surface area S and the cosine of the angle α between the normal \(~\vec n\) to the surface and the induction vector \(~\vec B\) (Fig. 1):

\(~\Phi = B \cdot S \cdot \cos \alpha .\) (1)

In SI, the unit of magnetic flux is weber(Wb):

1 Wb \u003d 1 T ⋅ 1 m 2.

• Magnetic flux at 1 Wb is the magnetic flux of a uniform magnetic field with an induction of 1 T through a flat surface perpendicular to it with an area of ​​1 m 2.

The flow can be either positive or negative depending on the value of the angle α. The flux of magnetic induction can be clearly interpreted as a quantity proportional to the number of lines of the induction vector \(~\vec B\) penetrating a given area of ​​the surface.

From formula (1) it follows that the magnetic flux can change:

• or only by changing the modulus of the induction vector B magnetic field, then \(~\Delta \Phi = (B_2 - B_1) \cdot S \cdot \cos \alpha\) ;
• or only by changing the contour area S, then \(~\Delta \Phi = B \cdot (S_2 - S_1) \cdot \cos \alpha\) ;
• or only due to contour rotation in a magnetic field, then \(~\Delta \Phi = B \cdot S \cdot (\cos \alpha_2 - \cos \alpha_1)\) ;
• or simultaneously by changing several parameters, then \(~\Delta \Phi = B_2 \cdot S_2 \cdot \cos \alpha_2 - B_1 \cdot S_1 \cdot \cos \alpha_1\) .

Electromagnetic induction (EMI)

Discovery of EMP

You already know that there is always a magnetic field around a current-carrying conductor. Is it possible, on the contrary, to create a current in the conductor with the help of a magnetic field? It was this question that interested the English physicist Michael Faraday, who in 1822 wrote in his diary: "Turn magnetism into electricity." And only after 9 years this problem was solved by him.

Opening electromagnetic induction, as Faraday called this phenomenon, was made on August 29, 1831. Initially, induction was discovered in conductors that were stationary relative to each other when the circuit was closed and opened. Then, clearly understanding that the approach or removal of current-carrying conductors should lead to the same result as closing and opening the circuit, Faraday proved through experiments that the current arises when the coils move relative to each other (Fig. 2).

On October 17, as recorded in his laboratory journal, an induction current was detected in the coil during the pushing in (or pulling out) of the magnet (Fig. 3).

Within one month, Faraday experimentally discovered that an electric current arises in a closed circuit with any change in the magnetic flux through it. The current obtained in this way is called induced current I i.

It is known that an electric current arises in the circuit when external forces act on free charges. The work of these forces when moving a single positive charge along a closed circuit is called the electromotive force. Therefore, when the magnetic flux changes through the surface bounded by the contour, external forces appear in it, the action of which is characterized by the EMF, which is called EMF induction and denoted by E i.

Induction current I i in the circuit and induction EMF E i are related by the following relation (Ohm's law):

\(~I_i = -\dfrac (E_i)(R),\)

where R- loop resistance.

• The phenomenon of the occurrence of induction EMF when the magnetic flux changes through the area bounded by the circuit is called phenomenon of electromagnetic induction. If the circuit is closed, then along with the induction EMF, an induction current also arises. James Clerk Maxwell proposed the following hypothesis: a changing magnetic field creates an electric field in the surrounding space, which leads the free charges into directed motion, i.e. creates an inductive current. The lines of force of such a field are closed, i.e. electric field vortex. Induction currents arising in massive conductors under the action of an alternating magnetic field are called Foucault currents or eddy currents.

Story

Here is a brief description of the first experiment, given by Faraday himself.

“On a wide wooden coil was wound a copper wire 203 feet long (a foot is equal to 304.8 mm), and between the turns of it was wound a wire of the same length, but isolated from the first cotton thread. One of these spirals was connected to a galvanometer, and the other to a strong battery, consisting of 100 pairs of plates ... When the circuit was closed, it was possible to notice a sudden, but extremely weak effect on the galvanometer, and the same was noticed when the current stopped. With the continuous passage of current through one of the coils, it was not possible to note any effect on the galvanometer, or in general any inductive effect on the other coil, despite the fact that the heating of the entire coil connected to the battery, and the brightness of the spark jumping between the coals, testified about battery power.

See also

1. Vasiliev A. Volta, Oersted, Faraday // Kvant. - 2000. - No. 5. - S. 16-17

Lenz's rule

Russian physicist Emiliy Lenz in 1833 formulated the rule ( Lenz's rule), which allows you to set the direction of the induction current in the circuit:

• the induction current arising in a closed circuit has such a direction in which its own magnetic flux created by it through the area bounded by the circuit tends to prevent the change in the external magnetic flux that caused this current.

• the inductive current has such a direction that it prevents the cause causing it.

For example, with an increase in the magnetic flux through the turns of the coil, the induction current has such a direction that the magnetic field it creates prevents the growth of the magnetic flux through the turns of the coil, i.e. the induction vector \((\vec(B))"\) of this field is directed against the induction vector \(\vec(B)\) of the external magnetic field. If the magnetic flux through the coil weakens, then the induction current creates a magnetic field with induction \ ((\vec(B))"\) which increases the magnetic flux through the turns of the coil.

See also

Law of EMP

Faraday's experiments showed that the induction emf (and the strength of the inductive current) in a conducting circuit is proportional to the rate of change of the magnetic flux. If in a short time Δ t the magnetic flux changes by ΔΦ, then the rate of change of the magnetic flux is equal to \(\dfrac(\Delta \Phi )(\Delta t)\). Taking into account the Lenz rule, D. Maxwell in 1873 gave the following formulation of the law of electromagnetic induction:

• The induction emf in a closed circuit is equal to the rate of change of the magnetic flux penetrating this circuit, taken with the opposite sign

\(~E_i = -\dfrac (\Delta \Phi)(\Delta t).\)

• This formula can only be applied with a uniform change in the magnetic flux.

• The minus sign in the law follows from Lenz's law. With an increase in the magnetic flux (ΔΦ > 0), the EMF is negative (E i < 0), т.е. индукционный ток имеет такое направление, что вектор магнитной индукции индукционного магнитного поля направлен против вектора магнитной индукции внешнего (изменяющегося) магнитного поля (рис. 4, а). При уменьшении магнитного потока (ΔΦ < 0), ЭДС положительная (Ei> 0) (Fig. 4b).

Rice. 4

In the International System of Units, the law of electromagnetic induction is used to establish the unit of magnetic flux. Since the induction emf E i expressed in volts, and time in seconds, then from the Weber EMP law can be determined as follows:

• the magnetic flux through the surface bounded by a closed loop is equal to 1 Wb, if, with a uniform decrease in this flux to zero in 1 s, an induction emf equal to 1 V occurs in the loop:

1 Wb \u003d 1 V ∙ 1 s.

EMF of induction in a moving conductor

When moving the conductor length l with a speed \(\vec(\upsilon)\) in a constant magnetic field with an induction vector \(\vec(B)\) an EMF of induction arises in it

\(~E_i = B \cdot \upsilon \cdot l \cdot \sin \alpha,\)

where α is the angle between the velocity direction \(\vec(\upsilon)\) of the conductor and the magnetic induction vector \(\vec(B)\).

The reason for the appearance of this EMF is the Lorentz force acting on free charges in a moving conductor. Therefore, the direction of the induced current in the conductor will coincide with the direction of the component of the Lorentz force on this conductor.

With this in mind, we can formulate the following to determine the direction of the induction current in a moving conductor ( left hand rule):

• you need to position your left hand so that the magnetic induction vector \(\vec(B)\) enters the palm, four fingers coincide with the direction of the velocity \(\vec(\upsilon)\) of the conductor, then the thumb set aside by 90 ° will indicate the direction induction current (Fig. 5).

If the conductor moves along the vector of magnetic induction, then there will be no inductive current (the Lorentz force is zero).

Literature

1. Aksenovich L. A. Physics in high school: Theory. Tasks. Tests: Proc. allowance for institutions providing general. environments, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Minsk: Adukatsia i vykhavanne, 2004. - C.344-351.
2. Zhilko V.V. Physics: textbook. allowance for the 11th grade. general education institutions with Russian. lang. Education with a 12-year term of study (basic and advanced levels) / V.V. Zhilko, L.G. Markovich. - Mn.: Nar. asveta, 2008. - S. 170-182.
3. Myakishev, G.Ya. Physics: Electrodynamics. 10-11 cells: textbook. for in-depth study of physics / G.Ya. Myakishev, A.3. Sinyakov, V.A. Slobodskov. - M.: Bustard, 2005. - S. 399-408, 412-414.

In this lesson, the topic of which is: “Lenz's rule. The law of electromagnetic induction ”, we will learn the general rule that allows us to determine the direction of the induction current in the circuit, established in 1833 by E.X. Lenz. We will also consider an experiment with aluminum rings, which clearly demonstrates this rule, and formulate the law of electromagnetic induction

By approaching or moving the magnet away from the solid ring, we change the magnetic flux that permeates the area of ​​the ring. According to the theory of the phenomenon of electromagnetic induction, an inductive electric current must occur in the ring. From the experiments of Ampere it is known that where the current passes, a magnetic field arises. Consequently, the closed ring begins to behave like a magnet. That is, there is an interaction of two magnets (a permanent magnet, which we move, and a closed circuit with current).

Since the system did not respond to the approach of the magnet to the ring with a cut, it can be concluded that the induction current does not occur in an open circuit.

Causes of repulsion or attraction of the ring to the magnet

1. When the magnet approaches

When the pole of the magnet approaches, the ring repels from it. That is, it behaves like a magnet, which has the same pole on our side as the approaching magnet. If we approach the north pole of the magnet, then the magnetic induction vector of the ring with inductive current is directed in the opposite direction relative to the magnetic induction vector of the north pole of the magnet (see Fig. 2).

Rice. 2. Approach of the magnet to the ring

2. When removing the magnet from the ring

When the magnet is removed, the ring trails behind it. Consequently, from the side of the receding magnet, the opposite pole is formed near the ring. The magnetic induction vector of the ring with current is directed in the same direction as the magnetic induction vector of the receding magnet (see Fig. 3).

Rice. 3. Removing the magnet from the ring

From this experiment, we can conclude that when the magnet moves, the ring also behaves like a magnet, the polarity of which depends on whether the magnetic flux penetrating the ring area increases or decreases. If the flux increases, then the magnetic induction vectors of the ring and the magnet are opposite in direction. If the magnetic flux through the ring decreases with time, then the magnetic field induction vector of the ring coincides in direction with the magnet induction vector.

The direction of the induced current in the ring can be determined by the right hand rule. If you point the thumb of your right hand in the direction of the magnetic induction vector, then four bent fingers will indicate the direction of the current in the ring (see Fig. 4).

Rice. 4. Right hand rule

When the magnetic flux penetrating the circuit changes, an induction current arises in the circuit in such a direction as to compensate for the change in the external magnetic flux with its magnetic flux.

If the external magnetic flux increases, then the induction current tends to slow down this increase with its magnetic field. If the magnetic flux decreases, then the inductive current with its magnetic field tends to slow down this decrease.

This feature of electromagnetic induction is expressed by the minus sign in the induction EMF formula.

Law of electromagnetic induction

When the external magnetic flux penetrating the circuit changes, an induction current appears in the circuit. In this case, the value of the electromotive force is numerically equal to the rate of change of the magnetic flux, taken with the "-" sign.

Lenz's rule is a consequence of the law of conservation of energy in electromagnetic phenomena.

Bibliography

1. Myakishev G.Ya. Physics: Proc. for 11 cells. general education institutions. - M.: Education, 2010.
2. Kasyanov V.A. Physics. Grade 11: Proc. for general education institutions. - M.: Bustard, 2005.
3. Gendenstein L.E., Dick Yu.I., Physics 11. - M .: Mnemosyne.

Homework

1. Questions at the end of paragraph 10 (p. 33) - Myakishev G.Ya. Physics 11 (see list of recommended reading)
2. How is the law of electromagnetic induction formulated?
3. Why is there a "-" sign in the formula for the law of electromagnetic induction?

1. Internet portal Festival.1september.ru ().
2. Internet portal Physics.kgsu.ru ().
3. Internet portal Youtube.com ().

Empirically, M. Faraday showed that the strength of the induction current in a conducting circuit is directly proportional to the rate of change in the number of magnetic induction lines that pass through the surface limited by the circuit in question. The modern formulation of the law of electromagnetic induction, using the concept of magnetic flux, was given by Maxwell. The magnetic flux (Ф) through the surface S is a value equal to:

where is the modulus of the magnetic induction vector; - the angle between the magnetic induction vector and the normal to the contour plane. The magnetic flux is interpreted as a quantity that is proportional to the number of magnetic induction lines passing through the considered surface area S.

The appearance of an induction current indicates that a certain electromotive force (EMF) arises in the conductor. The reason for the appearance of EMF induction is a change in the magnetic flux. In the system of international units (SI), the law of electromagnetic induction is written as follows:

where is the rate of change of the magnetic flux through the area that the contour limits.

The sign of the magnetic flux depends on the choice of the positive normal to the contour plane. In this case, the direction of the normal is determined using the rule of the right screw, connecting it with the positive direction of the current in the circuit. So, the positive direction of the normal is arbitrarily assigned, the positive direction of the current and the EMF of induction in the circuit are determined. The minus sign in the basic law of electromagnetic induction corresponds to Lenz's rule.

Figure 1 shows a closed loop. Assume that the positive direction of the contour traversal is counterclockwise, then the normal to the contour () is the right screw in the direction of traversal of the contour. If the magnetic induction vector of the external field is co-directed with the normal and its modulus increases with time, then we get:

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In this case, the induction current will create a magnetic flux (F '), which will be less than zero. The lines of magnetic induction of the magnetic field of the induction current () are shown in fig. 1 dotted line. The induction current will be directed clockwise. The induction emf will be less than zero.

Formula (2) is a record of the law of electromagnetic induction in the most general form. It can be applied to fixed circuits and conductors moving in a magnetic field. The derivative, which is included in expression (2), generally consists of two parts: one depends on the change in the magnetic flux over time, the other is associated with the movement (deformations) of the conductor in a magnetic field.

In the event that the magnetic flux changes in equal time intervals by the same amount, then the law of electromagnetic induction is written as:

If a circuit consisting of N turns is considered in an alternating magnetic field, then the law of electromagnetic induction will take the form:

where the quantity is called flux linkage.

Examples of problem solving

EXAMPLE 1

Exercise	What is the rate of change of the magnetic flux in the solenoid, which has N = 1000 turns, if an induction EMF equal to 200 V is excited in it?
Decision	The basis for solving this problem is the law of electromagnetic induction in the form: where is the rate of change of the magnetic flux in the solenoid. Therefore, we find the desired value as: Let's do the calculations:
Answer

EXAMPLE 2

Exercise	A square conducting frame is in a magnetic field that changes according to the law: (where and are constants). The normal to the frame makes an angle with the direction of the field magnetic induction vector. frame wall b. Get an expression for the instantaneous value of the induction emf ().
Decision	Let's make a drawing. As a basis for solving the problem, we take the basic law of electromagnetic induction in the form: