What could be better than reading about the basics on a Monday evening electrodynamics. That's right, you can find a lot of things that will be better. However, we still invite you to read this article. It does not take much time, and useful information will remain in the subconscious. For example, in an exam, under stress, it will be possible to successfully extract Faraday's law from the depths of memory. Since there are several Faraday laws, let's clarify that here we are talking about Faraday's law of induction.

Electrodynamics- a branch of physics that studies the electromagnetic field in all its manifestations.

This is the interaction of electric and magnetic fields, electric current, electromagnetic radiation, the influence of the field on charged bodies.

Here we do not aim to consider the whole of electrodynamics. God save! Let's take a closer look at one of its basic laws, which is called Faraday's law of electromagnetic induction.

History and definition

Faraday, in parallel with Henry, discovered the phenomenon of electromagnetic induction in 1831. True, I managed to publish the results earlier. Faraday's law is widely used in engineering, in electric motors, transformers, generators and chokes. What is the essence of Faraday's law for electromagnetic induction, to put it simply? And here's what!

When the magnetic flux changes through a closed conducting circuit, an electric current appears in the circuit. That is, if we twist a frame from wire and place it in a changing magnetic field (we take a magnet and twist it around the frame), current will flow through the frame!

This current Faraday called induction, and the phenomenon itself was called electromagnetic induction.

Electromagnetic induction- the occurrence of an electric current in a closed circuit when the magnetic flux passing through the circuit changes.

The formulation of the basic law of electrodynamics - Faraday's law of electromagnetic induction, looks and sounds as follows:

EMF, arising in the circuit, is proportional to the rate of change of the magnetic flux F through the loop.

And where does the minus come from, you ask. To explain the minus sign in this formula, there is a special Lenz's rule. It says that the minus sign, in this case, indicates how the emerging EMF is directed. The fact is that the magnetic field created by the induction current is directed in such a way that it prevents a change in the magnetic flux that caused the induction current.

Examples of problem solving

That seems to be all. The significance of Faraday's law is fundamental, because the basis of almost the entire electrical industry is built on the use of this law. To understand it faster, consider an example of solving a problem on Faraday's law.

And remember, friends! If the task is stuck like a bone in the throat, and there is no more strength to endure it - contact our authors! Now you know . We will quickly provide a detailed solution and clarify all questions!

As a result of numerous experiments, Faraday established the basic quantitative law of electromagnetic induction. He showed that whenever there is a change in the flux of magnetic induction coupled to the circuit, an induction current appears in the circuit. The occurrence of an inductive current indicates the presence of an electromotive force in the circuit, called the electromotive force of electromagnetic induction. Faraday found that the value of the EMF of electromagnetic induction E i is proportional to the rate of change of the magnetic flux:

E i \u003d -K, (27.1)

where K is the coefficient of proportionality, depending only on the choice of units of measurement.

In the SI system of units, the coefficient K = 1, i.e.

E i = - . (27.2)

This formula is Faraday's law of electromagnetic induction. The minus sign in this formula corresponds to the rule (law) of Lenz.

Faraday's law can also be formulated in this way: EMF of electromagnetic induction E i in the circuit is numerically equal and opposite in sign of the rate of change of the magnetic flux through the surface bounded by this circuit. This law is universal: EMF E i does not depend on how the magnetic flux changes.

The minus sign in (27.2) shows that an increase in flux (> 0) causes an EMF E i< 0, т.е. магнитный поток индукционного тока направлен навстречу потоку, вызвавшему его; уменьшение потока ( < 0) вызывает E i >0 i.e., the directions of the magnetic flux of the induction current and the flux that caused it are the same. The minus sign in formula (27.2) is a mathematical expression of Lenz's rule - a general rule for finding the direction of the induction current (and hence the sign and EMF of induction), derived in 1833. Lenz's rule: the induction current is always directed in such a way as to counteract the cause that causes it . In other words, the induction current creates a magnetic flux that prevents a change in the magnetic flux that causes the induction EMF.

The induction emf is expressed in volts (V). Indeed, given that the unit of magnetic flux is weber (Wb), we get:

If the closed circuit in which the induction EMF is induced consists of N turns, then E i will be equal to the sum of the EMF induced in each of the turns. And if the magnetic flux covered by each turn is the same and equal to Ф, then the total flux through the surface of N turns is equal to (NF) - the total magnetic flux (flux linkage). In this case, the induction emf is equal to:

E i = -N× , (27.3)

Formula (27.2) expresses the law of electromagnetic induction in a general form. It is applicable to both stationary circuits and moving conductors in a magnetic field. The time derivative of the magnetic flux included in it generally consists of two parts, one of which is due to the change in magnetic induction over time, and the other is due to the movement of the circuit relative to the magnetic field (or its deformation). Consider some examples of the application of this law.

Example 1. A straight conductor of length l moves parallel to itself in a uniform magnetic field (Figure 38). This conductor may be part of a closed circuit, the remaining parts of which are motionless. Find the EMF that occurs in the conductor.

If the instantaneous value of the conductor's speed is v, then in time dt he will describe the area dS = l× v×dt and during this time will cross all the lines of magnetic induction passing through dS. Therefore, the change in the magnetic flux through the circuit, which includes a moving conductor, will be dФ = B n ×l× v×dt. Here B n is the magnetic induction component perpendicular to dS. Substituting this into formula (27.2) we obtain the value of the EMF:

E i = B n×l× v. (27.4)

The direction of the induction current and the sign of the EMF are determined by the Lenz rule: the induction current in the circuit always has such a direction that the magnetic field it creates prevents a change in the magnetic flux that caused this induction current. In some cases, it is possible to determine the direction of the induction current (the polarity of the induction EMF) according to another formulation of the Lenz rule: the induction current in a moving conductor is directed in such a way that the resulting Ampère force is opposite to the velocity vector (slows down the movement).

Let's take a numerical example. A vertical conductor (car antenna) with a length l = 2 m moves from east to west in the Earth's magnetic field with a speed v= 72 km/h = 20 m/s. Calculate the voltage between the ends of the conductor. Since the conductor is open, there will be no current in it and the voltage at the ends will be equal to the induction emf. Considering that the horizontal component of the magnetic induction of the Earth's field (i.e., the component perpendicular to the direction of movement) for mid-latitudes is 2 × 10 -5 T, according to formula (27.4) we find

U = B n×l× v\u003d 2 × 10 -5 × 2 × 20 \u003d 0.8 × 10 -3 V,

those. about 1 mV. The Earth's magnetic field is directed from south to north. Therefore, we find that the EMF is directed from top to bottom. This means that the lower end of the wire will have a higher potential (will be charged positively), and the upper end will be lower (will be charged negatively).

Example 2. There is a closed wire circuit in a magnetic field, penetrated by a magnetic flux F. Let us assume that this flux decreases to zero, and calculate the total amount of charge that has passed through the circuit. The instantaneous value of the EMF in the process of the disappearance of the magnetic flux is expressed by the formula (27.2). Therefore, according to Ohm's law, the instantaneous value of the current strength is

where R is the impedance of the circuit.

The value of the passed charge is equal to

q = = - = . (27.6)

The resulting ratio expresses the law of electromagnetic induction in the form found by Faraday, who concluded from his experiments that the amount of charge passed through the circuit is proportional to the total number of magnetic induction lines crossed by the conductor (i.e., the change in the magnetic flux Ф 1 -Ф 2), and is inversely proportional to the resistance of the circuit R. Relation (27.6) allows us to define the unit of magnetic flux in the SI system: weber is a magnetic flux, when it decreases to zero, a charge of 1 C passes in a circuit with a resistance of 1 Ohm linked to it.

According to Faraday's law, the occurrence of electromagnetic induction EMF is also possible in the case of a fixed circuit located in an alternating magnetic field. However, the Lorentz force does not act on stationary charges, therefore, in this case, it cannot be the cause of the induction EMF. Maxwell, to explain the EMF of induction in stationary conductors, suggested that any alternating magnetic field excites a vortex electric field in the surrounding space, which is the cause of the induction current in the conductor. The circulation of the intensity vector of this field along any fixed circuit L of the conductor is the EMF of electromagnetic induction:

E i = = - . (27.7)

The lines of intensity of the vortex electric field are closed curves, therefore, when a charge moves in a vortex electric field along a closed circuit, non-zero work is performed. This is the difference between the vortex electric field and the electrostatic field, the lines of intensity of which begin and end on the charges.

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.

Faraday's law of electromagnetic induction.

We examined in sufficient detail three different, at first glance, variants of the phenomenon of electromagnetic induction, the occurrence of electric current in a conducting circuit under the influence of a magnetic field: when the conductor moves in a constant magnetic field; when the source of the magnetic field moves; when the magnetic field changes with time. In all these cases, the law of electromagnetic induction is the same:
The EMF of electromagnetic induction in the circuit is equal to the rate of change of the magnetic flux through the circuit, taken with the opposite sign

regardless of the reasons leading to a change in this flow.
Let us clarify some details of the above formulation.
 First. The magnetic flux through the circuit can vary arbitrarily, that is, the function Ф(t) does not have to always be linear, but can be any. If the magnetic flux changes linearly, then the induction EMF in the circuit is constant, in this case the value of the time interval Δt can be arbitrary, the value of relation (1) in this case does not depend on the value of this interval. If the flow changes in a more complex way, then the value of the EMF is not constant, but depends on time. In this case, the time interval under consideration should be considered infinitely small, then relation (1) from a mathematical point of view turns into a derivative of the magnetic flux function with respect to time. Mathematically, this transition is completely analogous to the transition from average to instantaneous speed in kinematics.
 Second. The concept of the flow of a vector field is applicable only to a surface, so it is necessary to specify what kind of surface is meant in the formulation of the law. However, the magnetic field flux through any closed surface is zero. Therefore, for two different surfaces based on the contour, the magnetic fluxes are the same. Imagine a stream of fluid flowing out of a hole. Whichever surface you choose, the boundaries of which are the boundaries of the hole, the flows through them will be the same. Another analogy is appropriate here: if the work of a force along a closed contour is zero, then the work of this force does not depend on the shape of the trajectory, but is determined only by its initial and final points.
 Third. The minus sign in the formulation of the law has a deep physical meaning; in fact, it ensures the fulfillment of the energy conservation law in these phenomena. This sign is an expression of Lenz's rule. Perhaps this is the only case in physics when one sign was awarded its own name.
As we have shown, in all cases the physical essence of the phenomenon of electromagnetic induction is the same and is briefly formulated as follows: an alternating magnetic field generates a vortex electric field. From this, field, point of view, the law of electromagnetic induction is expressed through the characteristics of the electromagnetic field: the circulation of the electric field strength vector along any circuit is equal to the rate of change of the magnetic flux through this circuit

In this interpretation of the phenomenon, it is essential that a vortex electric field arises when the magnetic field changes, regardless of whether there is a real closed conductor (circuit) in which current occurs or not. This real circuit can play the role of a device to detect the induced field.
Finally, we emphasize once again that electric and magnetic fields are relative, that is, their characteristics depend on the choice of the reference frame in which they are described. However, this arbitrariness in the choice of reference system, in the choice of the method of description does not lead to any contradictions. The measured physical quantities are invariant and do not depend on the choice of reference system. For example, the force acting on a charged body from the side of an electromagnetic field does not depend on the choice of the frame of reference. But when it is described in some systems, it can be interpreted as the Lorentz force, in others, an electric force can be "added" to it. Similarly (even as a consequence), the EMF of induction in the circuit (the strength of the induced current, the amount of heat released, the possible deformation of the circuit, etc.) do not depend on the choice of the reference system.
As always, the given freedom of choice can and should be used - there is always the opportunity to choose the method of description that you like best - as the simplest, most visual, most familiar, etc.

Phenomenon electromagnetic induction was discovered by an eminent English physicist M. Faraday in 1831. It consists in the occurrence of an electric current in a closed conducting circuit with a change in time magnetic flux penetrating the contour.

Magnetic flux Φ through the area S the contour is called the value

where B– module magnetic induction vector, α is the angle between the vector and the normal to the contour plane (Fig. 1.20.1).

The definition of magnetic flux can be easily generalized to the case of an inhomogeneous magnetic field and a non-planar contour. The unit of magnetic flux in the SI system is called weber (Wb). A magnetic flux equal to 1 Wb is created by a magnetic field with an induction of 1 T, penetrating a flat contour with an area of ​​1 m 2 in the direction of the normal:

Faraday experimentally established that when the magnetic flux changes in a conducting circuit, an induction emf ind arises, equal to the rate of change of the magnetic flux through the surface bounded by the circuit, taken with a minus sign:

This formula is called Faraday's law .

Experience shows that the induction current excited in a closed circuit when the magnetic flux changes is always directed in such a way that the magnetic field it creates prevents a change in the magnetic flux that causes the inductive current. This statement, formulated in 1833, is called Lenz's rule .

Rice. 1.20.2 illustrates Lenz's rule using the example of a fixed conducting circuit, which is in a uniform magnetic field, the modulus of induction of which increases with time.

Lenz's rule reflects the experimental fact that ind and always have opposite signs (the minus sign in Faraday's formula). Lenz's rule has a deep physical meaning - it expresses the law of conservation of energy.

A change in the magnetic flux penetrating a closed circuit can occur for two reasons.

1. The magnetic flux changes due to the movement of the circuit or its parts in a magnetic field constant in time. This is the case when conductors, and with them free charge carriers, move in a magnetic field. The occurrence of the induction EMF is explained by the action of the Lorentz force on free charges in moving conductors. Lorentz force plays the role of an external force in this case.

Consider, as an example, the occurrence of induction EMF in a rectangular circuit placed in a uniform magnetic field perpendicular to the plane of the circuit. Let one of the sides of the contour be l slides with speed along the other two sides (Fig. 1.20.3).

Lorentz force acts on free charges in this section of the contour. One of the components of this force, associated with portable the speed of the charges is directed along the conductor. This component is shown in Fig. 1.20.3. She plays the role of an external force. Its modulus is

According to the definition of EMF

In order to set the sign in the formula connecting ind and it is necessary to choose the direction of the normal and the positive direction of the contour traversal, which are consistent with each other according to the rule of the right gimlet, as is done in Fig. 1.20.1 and 1.20.2. If this is done, then it is easy to come to the Faraday formula.

If the resistance of the entire circuit is R, then an inductive current will flow through it, equal to I ind = ind / R. During the time Δ t on resistance R stand out joule heat

The question arises: where does this energy come from, because the Lorentz force does no work! This paradox arose because we took into account the work of only one component of the Lorentz force. When an inductive current flows through a conductor in a magnetic field, free charges are affected by another component of the Lorentz force, associated with relative the speed of charge movement along the conductor. This component is responsible for the appearance Ampere forces. For the case shown in Fig. 1.20.3, the Ampere force modulus is F A= I B l. The Ampere force is directed towards the movement of the conductor; therefore, it performs negative mechanical work. During the time Δ t this work A fur is

A conductor moving in a magnetic field, through which an induction current flows, experiences magnetic braking . The total work of the Lorentz force is zero. Joule heat is released in the circuit either due to the work of an external force that keeps the speed of the conductor unchanged, or due to a decrease in the kinetic energy of the conductor.

2. The second reason for the change in the magnetic flux penetrating the circuit is the change in time of the magnetic field when the circuit is stationary. In this case, the occurrence of the induction EMF can no longer be explained by the action of the Lorentz force. Electrons in a fixed conductor can only be set in motion by an electric field. This electric field is generated by a time-varying magnetic field. The work of this field when moving a single positive charge along a closed circuit is equal to the induction EMF in a stationary conductor. Therefore, the electric field generated by the changing magnetic field, is not potential . He's called vortex electric field . The concept of a vortex electric field was introduced into physics by the great English physicist J. Maxwell in 1861

The phenomenon of electromagnetic induction in fixed conductors, which occurs when the surrounding magnetic field changes, is also described by the Faraday formula. Thus, the phenomena of induction in moving and stationary conductors proceed the same way, but the physical cause of the induction current is different in these two cases: in the case of moving conductors, the induction EMF is due to the Lorentz force; in the case of fixed conductors, the induction EMF is a consequence of the action on free charges of the vortex electric field that occurs when the magnetic field changes.

If there is a closed conducting circuit in the magnetic field that does not contain current sources, then when the magnetic field changes, an electric current arises in the circuit. This phenomenon is called electromagnetic induction. The appearance of a current indicates the occurrence of an electric field in the circuit, which can provide a closed movement of electric charges or, in other words, the occurrence of an EMF. The electric field, which arises when the magnetic field changes and whose work is not equal to zero when moving charges along a closed circuit, has closed lines of force and is called vortex.

For a quantitative description of electromagnetic induction, the concept of magnetic flux (or magnetic induction vector flux) through a closed loop is introduced. For a flat circuit located in a uniform magnetic field (and only such situations can be encountered by schoolchildren at a unified state exam), the magnetic flux is defined as

where is the field induction, is the contour area, is the angle between the induction vector and the normal (perpendicular) to the contour plane (see figure; the perpendicular to the contour plane is shown by a dotted line). The unit of magnetic flux in the international SI system of units is Weber (Wb), which is defined as the magnetic flux through a contour of area 1 m 2 of a uniform magnetic field with an induction of 1 T, perpendicular to the plane of the contour.

The value of the EMF of induction that occurs in the circuit when the magnetic flux through this circuit changes is equal to the rate of change of the magnetic flux

Here is the change in the magnetic flux through the circuit over a small time interval. An important property of the law of electromagnetic induction (23.2) is its universality with respect to the reasons for changing the magnetic flux: the magnetic flux through the circuit can change due to a change in the magnetic field induction, a change in the area of ​​the circuit, or a change in the angle between the induction vector and the normal, which occurs when the circuit rotates in field. In all these cases, according to the law (23.2), induction EMF and induction current will appear in the circuit.

The minus sign in formula (23.2) is "responsible" for the direction of the current resulting from electromagnetic induction (Lenz's rule). However, it is not so easy to understand in the language of the law (23.2) which direction of the induction current this sign will lead to with this or that change in the magnetic flux through the circuit. But it is easy enough to remember the result: the induction current will be directed in such a way that the magnetic field created by it will “tend” to compensate for the change in the external magnetic field that generated this current. For example, with an increase in the flow of an external magnetic field through a circuit, an induction current will appear in it, the magnetic field of which will be directed opposite to the external magnetic field so as to reduce the external field and thus maintain the original value of the magnetic field. With a decrease in the field flux through the circuit, the induction current field will be directed in the same way as the external magnetic field.

If the current in a circuit with current changes for some reason, then the magnetic flux through the circuit of the magnetic field that is created by this current itself also changes. Then, according to the law (23.2), induction EMF should appear in the circuit. The phenomenon of the occurrence of an EMF of induction in a certain electrical circuit as a result of a change in current in this circuit itself is called self-induction. To find the EMF of self-induction in some electrical circuit, it is necessary to calculate the flux of the magnetic field created by this circuit through itself. Such a calculation is a difficult problem due to the inhomogeneity of the magnetic field. However, one property of this flow is obvious. Since the magnetic field created by the current in the circuit is proportional to the magnitude of the current, then the magnetic flux of the own field through the circuit is proportional to the current in this circuit

where is the current strength in the circuit, is the coefficient of proportionality, which characterizes the "geometry" of the circuit, but does not depend on the current in it and is called the inductance of this circuit. The unit of inductance in the international SI system of units is Henry (H). 1 H is defined as the inductance of such a circuit, the flux of induction of its own magnetic field through which is 1 Wb at a current strength of 1 A. Taking into account the definition of inductance (23.3) from the law of electromagnetic induction (23.2), we obtain for the EMF of self-induction

Due to the phenomenon of self-induction, the current in any electrical circuit has a certain "inertia" and, therefore, energy. Indeed, to create a current in the circuit, it is necessary to do work to overcome the self-induction EMF. The energy of the circuit with current and is equal to this work. It is necessary to remember the formula for the energy of the circuit with current

where is the inductance of the circuit, is the current in it.

The phenomenon of electromagnetic induction is widely used in technology. It is based on the creation of electric current in electric generators and power plants. Thanks to the law of electromagnetic induction, mechanical vibrations are converted into electrical vibrations in microphones. On the basis of the law of electromagnetic induction, in particular, an electric circuit, which is called an oscillatory circuit (see the next chapter), and which is the basis of any radio transmitting or radio receiving equipment, works.

Consider now the tasks.

Of those listed in task 23.1.1 phenomena, there is only one consequence of the law of electromagnetic induction - the appearance of a current in the ring when a permanent magnet is passed through it (the answer 3 ). Everything else is the result of the magnetic interaction of currents.

As indicated in the introduction to this chapter, the phenomenon of electromagnetic induction underlies the operation of an alternator ( task 23.1.2), i.e. device that creates alternating current, a given frequency (the answer 2 ).

The induction of the magnetic field created by a permanent magnet decreases with increasing distance from it. Therefore, when the magnet approaches the ring ( task 23.1.3) the induction flux of the magnetic field of the magnet through the ring changes, and an induction current appears in the ring. Obviously, this will happen when the magnet approaches the ring with both the north and south poles. But the direction of the induction current in these cases will be different. This is due to the fact that when the magnet approaches the ring with different poles, the field in the plane of the ring in one case will be directed opposite to the field in the other. Therefore, to compensate for these changes in the external field, the magnetic field of the inductive current must be directed differently in these cases. Therefore, the directions of the induction currents in the ring will be opposite (the answer is 4 ).

For the occurrence of an EMF of induction in the ring, it is necessary that the magnetic flux through the ring changes. And since the magnetic induction of the magnet field depends on the distance to it, then in the considered case task 23.1.4 case, the flow through the ring will change, an induction current will appear in the ring (the answer is 1 ).

When rotating frame 1 ( task 23.1.5) the angle between the lines of magnetic induction (and, therefore, the induction vector) and the plane of the frame at any time is equal to zero. Consequently, the magnetic flux through the frame 1 does not change (see formula (23.1)), and the induction current does not occur in it. In frame 2, an induction current will occur: in the position shown in the figure, the magnetic flux through it is zero, when the frame turns a quarter of a turn, it will be equal to , where is the induction, is the area of ​​the frame. After another quarter of a turn, the flow will again be zero, and so on. Therefore, the flux of magnetic induction through frame 2 changes during its rotation, therefore, an induction current arises in it (the answer is 2 ).

AT task 23.1.6 induction current occurs only in case 2 (answer 2 ). Indeed, in case 1, the frame remains at the same distance from the conductor during movement, and, consequently, the magnetic field created by this conductor in the plane of the frame does not change. When the frame moves away from the conductor, the magnetic induction of the conductor field in the frame area changes, the magnetic flux through the frame changes, and an induction current arises

The law of electromagnetic induction states that an inductive current will flow in a ring at times when the magnetic flux through this ring changes. Therefore, while the magnet is at rest near the ring ( task 23.1.7) the inductive current in the ring will not flow. So the correct answer for this problem is 2 .

According to the law of electromagnetic induction (23.2), the induction EMF in the frame is determined by the rate of change of the magnetic flux through it. And since by condition tasks 23.1.8 the induction of the magnetic field in the region of the frame changes uniformly, the rate of its change is constant, the magnitude of the induction emf does not change during the experiment (the answer is 3 ).

AT task 23.1.9 The induction emf that occurs in the frame in the second case is four times greater than the induction emf that occurs in the first (the answer is 4 ). This is due to a fourfold increase in the frame area and, accordingly, the magnetic flux through it in the second case.

AT task 23.1.10 in the second case, the rate of change of the magnetic flux doubles (the field induction changes by the same amount, but in half the time). Therefore, the EMF of electromagnetic induction that occurs in the frame in the second case is twice as large as in the first (the answer is 1 ).

When the current in a closed conductor doubles ( task 23.2.1), the value of the induction of the magnetic field will increase at each point in space twice, without changing in direction. Therefore, the magnetic flux through any small area and, accordingly, the entire conductor will change exactly twice (the answer is 1 ). But the ratio of the magnetic flux through the conductor to the current in this conductor, which is the inductance of the conductor , while not changing ( task 23.2.2- answer 3 ).

Using formula (23.3) we find in task 32.2.3 gn (answer 4 ).

Relationship between the units of measurement of magnetic flux, magnetic induction and inductance ( task 23.2.4) follows from the definition of inductance (23.3): a unit of magnetic flux (Wb) is equal to the product of a unit of current (A) per unit of inductance (H) - the answer 3 .

According to formula (23.5), with a twofold increase in the inductance of the coil and a twofold decrease in the current in it ( task 23.2.5) the energy of the magnetic field of the coil will decrease by 2 times (the answer 2 ).

When the frame rotates in a uniform magnetic field, the magnetic flux through the frame changes due to a change in the angle between the perpendicular to the plane of the frame and the magnetic field vector. And since in the first and second cases in task 23.2.6 this angle changes according to the same law (by condition, the frequency of rotation of the frames is the same), then the induction EMF changes according to the same law, and, therefore, the ratio of the amplitude values ​​of the induction EMF within the framework is equal to one (the answer 2 ).

The magnetic field created by a conductor with current in the region of the frame ( task 23.2.7), sent "from us" (see the solution of problems in Chapter 22). The value of the wire field induction in the frame area will decrease as it moves away from the wire. Therefore, the induction current in the frame must create a magnetic field directed inside the frame "away from us". Now using the gimlet rule to find the direction of magnetic induction, we conclude that the induction current in the loop will be directed clockwise (the answer is 1 ).

With an increase in current in the wire, the magnetic field created by it will increase and an induction current will appear in the frame ( task 23.2.8). As a result, there will be an interaction of the induction current in the loop and the current in the conductor. To find the direction of this interaction (attraction or repulsion), you can find the direction of the inductive current, and then, using the Ampère formula, the force of interaction between the frame and the wire. But you can do it differently, using the Lenz rule. All inductive phenomena must have such a direction as to compensate for the cause that causes them. And since the reason is an increase in the current in the loop, the force of interaction between the inductive current and the wire should tend to reduce the magnetic flux of the wire field through the loop. And since the magnetic induction of the wire field decreases with increasing distance to it, this force will repel the frame from the wire (answer 2 ). If the current in the wire decreased, then the frame would be attracted to the wire.

Task 23.2.9 also related to the direction of induction phenomena and Lenz's rule. When a magnet approaches a conducting ring, an induction current will appear in it, and its direction will be such as to compensate for the cause that causes it. And since this reason is the approach of a magnet, the ring will repel from it (answer 2 ). If the magnet is moved away from the ring, then for the same reasons there would be an attraction of the ring to the magnet.

Task 23.2.10 is the only computational problem in this chapter. To find the EMF of induction, you need to find the change in the magnetic flux through the circuit . It can be done like this. Let at some point in time the jumper was in the position shown in the figure, and let a small time interval pass. During this time interval, the jumper will move by the value . This will increase the contour area by the amount . Therefore, the change in the magnetic flux through the circuit will be equal, and the magnitude of the induction emf (answer 4 ).