In 1831, the English physicist M. Faraday discovered the phenomenon in his experiments electromagnetic induction. 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- this 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 - and this is wonderful. After all, before the discovery of this phenomenon, people did not know about methods of producing electric current, except for galvanization.
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:
But in the general case, the EMF formula with general flux linkage is used:
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 rules.
Lenz's rule
The current induced when the magnetic field passing through the circuit changes, its magnetic field prevents this change.
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.
The phenomenon of electromagnetic induction is that as a result of a change in time of the magnetic flux that penetrates a closed conducting circuit, an electric current arises in the circuit. This phenomenon was discovered by British physicist Max Faraday in 1831.
Let us introduce the notation we need to write the formula. To denote the magnetic flux we use the letter Ф, the contour area - S, the magnitude of the magnetic induction vector - B, α is the angle between the vector B → and the normal n → to the plane of the contour.
The magnetic flux that passes through the area of a closed conducting loop can be given by the following formula:
Φ = B S cos α,
Let's illustrate the formula.
Picture 1 . 20 . 1 . Magnetic flux through a closed loop. The normal direction n → and the selected positive direction l → bypassing the contour are related by the right gimlet rule.
The unit of magnetic flux in SI is taken to be 1 weber (V b). A magnetic flux equal to 1 V b can be created in a flat circuit with an area of 1 m 2 under the influence of a magnetic field with an induction of 1 T l, which penetrates the circuit in the normal direction.
1 V b = 1 T l m 2
Faraday's law
A change in the magnetic flux leads to the appearance of an induced emf δ and n in the conducting circuit. It is equal to the speed at which the magnetic flux changes through the surface limited by the circuit, taken with a minus sign. This was first experimentally established by Max Faraday. He wrote down his observation in the form of the induced emf formula, which is now called Faraday’s Law:
Definition 1
Faraday's Law:
δ and n d = - ∆ Φ ∆ t
Lenz's rule
Definition 2According to experimental results, the induction current that occurs in a closed loop as a result of a change in magnetic flux is always directed in a certain way. The magnetic field created by the induction current prevents the change in the magnetic flux that caused this induction current. Lenz formulated this rule in 1833.
Let us illustrate Lenz's rule with a drawing that depicts a stationary closed conducting circuit placed in a uniform magnetic field. The induction modulus increases with time.
Example 1
Thanks to Lenz's rule, we can justify the fact that in the formula of electromagnetic induction δ and n d and ∆ Φ ∆ t are opposite in sign.
If you think about the physical meaning of Lenz's rule, then this is a special case of the Law of Conservation of Energy.
There are two reasons why a change in the magnetic flux penetrating a closed circuit may occur:
- Change in magnetic flux due to movement of the entire circuit or its individual parts in a magnetic field that does not change over time;
- Change in magnetic field with a stationary circuit.
Let's move on to consider these cases in more detail.
Moving a circuit or its parts in a constant magnetic field
When conductors and free charge carriers move in a magnetic field, an induced emf occurs. The occurrence of δ and n d can be explained by the action of the Lorentz force on free charges in moving conductors. The Lorentz force here is an external force.
Example 2
In the figure we depicted an example of induction when a rectangular contour is placed in a uniform magnetic field B → directed perpendicular to the plane of the contour. One side of the contour moves along the other two sides at a certain speed.
Picture 1 . 20 . 3. The occurrence of induced emf in a moving conductor. The component of the Lorentz force that acts on a free electron is reflected
The free charges of the moving part of the circuit are affected by the Lorentz force. The main component of the Lorentz force in this case is directed along the conductor and is associated with the transfer velocity of charges υ →. The modulus of this external force is equal to:
F L = e υ → B.
The work done by force F L on path l is equal to:
A = F L · l = e υ B l .
According to the definition of EMF:
δ and n d = A e = υ B l .
The value of the external force for the stationary parts of the contour is zero. For the relationship between δ and n d, you can write another version of the formula. The contour area changes over time by Δ S = l υ Δ t. Accordingly, the magnetic flux will also change over time: Δ Φ = B l υ Δ t.
Hence,
δ and n d = ∆ Φ ∆ t.
The signs in the formula that relates δ and ind and ∆ Φ ∆ t can be set depending on which normal and contour directions are chosen. In the case of choosing the normal directions n → and the positive direction of traversing the contour l → consistent with each other according to the right-hand gimlet rule, one can arrive at Faraday’s formula.
Provided that the resistance of the entire circuit is R, then an induction current will flow through it, which is equal to I and n d = δ and n d R. During the time Δt at the resistance R Joule heat will be released:
∆ Q = R I and n d 2 ∆ t = υ 2 B 2 l 2 R ∆ t
There is no paradox here. We simply did not take into account the impact of another force on the system. The explanation is that when an induction current flows through a conductor located in a magnetic field, another component of the Lorentz force acts on the free charges, which is associated with the relative speed of movement of the charges along the conductor. Thanks to this component, the Ampere force F A → appears.
For the example considered above, the modulus of the Ampere force is equal to F A = I B l. The direction of the Ampere force is such that it performs negative mechanical work A me x. This mechanical work over a certain period of time can be calculated using the formula:
A me x = - F υ ∆ t = - I B l υ ∆ t = - υ 2 B 2 l 2 R ∆ t
A conductor moving in a magnetic field experiences magnetic braking. This leads to the fact that the total work done by the Lorentz force is zero. Joule heat can be released either due to a decrease in the kinetic energy of a moving conductor, or due to the energy that maintains the speed of movement of the conductor in space.
Change in magnetic field with a stationary circuit
Definition 3Vortex electric field is an electric field that is caused by a changing magnetic field.
In contrast to the potential electric field, the work of the vortex electric field when moving a single positive charge along a closed conducting circuit is equal to δ and n d in a stationary conductor.
In a stationary conductor, electrons can only be moved by an electric field. And the occurrence of δ and n d cannot be explained by the action of the Lorentz force.
The first to introduce the concept of a vortex electric field was the English physicist John Maxwell. This happened in 1861.
In fact, the phenomena of induction in moving and stationary conductors proceed in the same way. So in this case we can also use Faraday's formula. The differences relate to the physical cause of the occurrence of induced current: in moving conductors δ and n d is determined by the Lorentz force, in stationary ones - by the effect on free charges of the vortex electric field that arises when the magnetic field changes.
Definition 1
E.H. Lenz offered rule (law), which allows you to find direction of induction current. In its formulation it is as follows: “If a metal conductor moves near a galvanic current or near a magnet, then a galvanic current of such a direction is excited in it that would cause the motion of the resting wire in the direction directly opposite to the direction of motion imposed here on the wire from the outside, on the assumption that a wire at rest can only move in the direction of this last movement or in the exact opposite direction.”
Definition 2
Currently, Lenz’s rule is formulated more briefly: “The direction of the induced current is such that its effect is opposite to the action of the cause that causes it.” Or: The induction currents that appear in a conductor as a result of their movement in a constant magnetic field have a direction in which the pondemotive forces of the magnetic field that these conductors experience impede the movement of the conductors.
This rule is observed in all cases of induction.
Picture 1.
Let us assume that induction occurs in circuit (2) when it moves in the magnetic field of a circuit with current (1) (Fig. 1). In this case, an induction current appears, having such a direction that the force of interaction with the circuit (1) counteracts the movement of the circuit. If circuit (2) is brought closer to circuit (1), a current $I_2"$ appears, and the magnetic moment of this current is directed opposite the current field $I_1$. Circuit (2) is subject to a force that pushes it away from circuit (1). If circuit (2) is removed from circuit (1), a current $I^("")_2 will appear in circuit (2), the direction of its moment will coincide with the current field $I_1$, therefore, the force that acts on circuit (2) attracts it to the circuit (1).
Let us assume that both circuits are stationary; an alternating current $I_1$ flows in circuit (1), changes in which are caused by the appearance of current $I_2$. The direction of the current in the second duct is such that the magnetic flux $(Ф)$ created by this current tends to weaken changes in the external flux, which leads to the emergence of an induction current. As the current $I_1$ increases, the external magnetic flux, which is directed to the right, increases, and a current $I_2"$ appears, which creates a flux directed to the left (Fig. 1).
If the current $I_1$ decreases, a current $I^("")_2 appears in circuit (2), the magnetic flux of which is directed in the same way as the external flux; the additional magnetic flux maintains the external flux unchanged.
Lenz's rule and the law of conservation of energy
Lenz's law is a consequence of the law of conservation of energy. Induction currents, like any other, produce work. For example, if a closed conductor moves in a magnetic field, additional work must be done by external forces because the induced currents interact with the magnetic field, generating forces that are directed opposite to the movement.
Example 1
Exercise: Indicate the direction of the induction current that appears in the circuit a) if the magnet is brought closer to the circuit; b) when the magnet is removed from the circuit (Fig. 2). Explain how a magnet and a current-carrying coil interact in cases a) and b).
Figure 2.
Solution:
When we bring the north pole of the magnet $(N)$ closer to the contour, then a north magnetic pole also appears on the contour. When we remove the north pole of the magnet from the circuit, a south pole appears on the circuit. In this case, like poles of a magnet repel, and unlike poles attract. This means that when an induced current appears in the circuit when the magnet approaches the circuit, the interaction forces between the magnet and the induced current repel the magnet from the coil, and if a current arises in the circuit when the magnet is removed, the coil with the induced current and the magnet are attracted.
In accordance with Lenz's rule, the directions of currents will have the directions indicated in Fig. 3.
Figure 3.
Example 2
Exercise: A straight conductor of length $l$ moves parallel to itself in a magnetic field. This conductor can be part of a closed circuit, the remaining parts of which are motionless. Find the EMF that occurs in the conductor, indicate the direction of the induced current.
Solution:
Figure 4.
Let us denote by $v$ the instantaneous speed of movement of the conductor, $dt$ - the time of movement of the conductor, then the conductor will describe an area equal to:
In time $dt$, the conductor will cross all lines of magnetic induction that pass through the area $dS$. The change in magnetic flux can therefore be written as:
where $B_n$ is the component of magnetic induction, which is perpendicular to the area $dS$. Using Faraday's law, we get:
\[((\mathcal E))_i=-\frac(dФ)(dt)=(-B)_nlv.\]
The direction of the induced current and the sign of the emf are determined by Lenz's rule. The current is directed so that the mechanical force acting on the conductor is opposite to the speed.
Answer:$((\mathcal E))_i=(-B)_nlv.$
Lenz's rule or law received its name in honor of the German-born physicist who lived and taught in Russia, Emilius Lenz. His rule obeys Newton’s third law (for every action there is an equal reaction) and the law of conservation of energy (in a closed system, energy can neither appear nor disappear, so the sum of all energies in it remains constant).
Lenz's rule is based on Faraday's law of electromagnetic induction. It is necessary to remember that an external alternating magnetic field acting on the coil causes an EMF in it.
Moving a permanent magnet toward or away from the coil changes the magnetic flux passing through the coil circuit. The magnitude of the EMF induced in the circuit is directly proportional to the rate of change of the magnetic flux.
In situations a) and c), when the magnet is brought closer to the coil or moved away from it, electrons begin to move in a directional manner in the coil (current is induced). In situation b) the magnet is stationary, therefore, we can say that the magnetic field is constant and there is no current in the coil.
How do you know where the induced current is directed?
Emilius Lenz formulated a simple rule (law) that explains the direction of the current induced in the coil:
The induced current flows so as to counteract with its magnetic field the changing flux of the external magnetic field by which it is caused.
Lenz's rule explained
To understand Lenz's law, let us pay attention to two experimental situations.
The magnet approaches the coil
They tend to bring the north pole of the magnet closer to the coil. The magnetic flux passing through the turns of the coil increases. The current appearing in the coil creates a magnetic field around it. According to Lenz's rule, it opposes the increase in magnetic flux through the coil. This situation is possible only when the side of the coil closest to the magnet acquires the polarity of the north pole. Knowing the polarity, you can easily determine the direction of the induced current by applying the right-hand rule. The current flows in a counterclockwise direction.
The magnet moves away from the coil
When the north pole of a magnet moves away from the coil, the magnetic flux through the coil decreases. A current arises in the coil according to Faraday's law. This current creates its own magnetic field. According to Lenz's rule, this magnetic field will oppose the decrease in magnetic flux through the coil. This is only possible if there is a south magnetic pole on the side of the coil closest to the magnet. Opposite poles attract. We know the polarity of the coil. Let's apply the right-hand rule and determine the direction of the current in the coil. In this situation it flows clockwise.
