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Description
Electromagnetic induction is a phenomenon consisting of the appearance of an electromotive force (induction emf) in a conductive circuit with any change in the magnetic flux crossing it.
The reasons for the change in magnetic flux can be both a change in time of magnetic induction created by external sources in a stationary circuit of unchanged shape and size, and changes in time of the position, shape and size of the circuit itself located in a magnetic field.
In accordance with Faraday's law (established independently by D. Henry and M. Faraday in 1831), the induced emf E in a circuit is directly proportional to the rate of change in time t of the magnetic flux F passing through the surface S bounded by the circuit, i.e.
E= - dФ/dt.
The minus sign determines the direction of the induced current in a closed loop, i.e. the induced current in the circuit is directed in such a way that the magnetic flux it creates through the surface bounded by this circuit prevents the change in flux Ф that caused the appearance of this current.
In a constant magnetic field, an induced emf occurs only if the conducting circuit moves non-colinear to the magnetic field lines or changes its shape and size over time.
Illustration of the occurrence of induced emf in a moving conducting frame
Rice. 1
If a rectilinear element of length l of a conducting circuit (see Fig. 1) moves with a constant speed V at an angle a to the direction of the lines of force of a constant magnetic field with induction B, then the magnetic flux over a period of time dt will change by the amount:
dФ=(Вldx)sin a.
The induced emf will be:
E= - BlVsin a.
The phenomenon of electromagnetic induction manifests itself in a closed conductor of any geometric shape.
The induced emf is numerically equal to the work of moving a unit charge along a closed loop, performed by the forces of the vortex electric field, which is generated in space when the magnetic field changes over time.
Timing characteristics
Initiation time (log to -6 to -3);
Lifetime (log tc from -3 to 9);
Degradation time (log td from -6 to -3);
Time of optimal development (log tk from -1 to 7).
Diagram:
Technical implementations of the effect
Technical implementation of the effect
The simplest technical implementation is shown in Fig. 2.
Scheme of the simplest device for observing induced emf
Rice. 2
Designations:
1 - coil;
2 - winding;
3 - permanent magnet;
4 - magnet support;
5 - device for measuring induced emf.
A permanent magnet is introduced instead of the induction coil core. When the magnet is removed, an EMF pulse occurs, the amplitude of which is proportional to the speed of removal of the magnet.
Applying an effect
Example 11.7.
Magnetic flux through a closed conducting loop with resistance R= 10 ohms changes over time t according to the law Ф = t 2, where = 10 Wb/s 2. Determine the current strength I in the circuit at the moment of time t= 1 ms.
Solution.
The instantaneous value of the induced emf, according to Faraday's law, is determined as
Then the current in the circuit according to Ohm’s law is equal to
mA.
The minus sign in the resulting expression indicates that the direction of the induction current is opposite to the direction of the positive circuit bypass, which in turn is consistent with the direction of the normal vector to the surface stretched over the circuit. The cause of the induction current is the vortex electric field generated by the changing magnetic field if the circuit is stationary, and the Lorentz force if it moves in a non-uniform constant magnetic field.
Example 11.8.
On a long solenoid having a cross-sectional diameter d= 5 cm and containing n= 20 turns per 1 cm of length, a circular turn of copper wire with a section of s= 1 mm 2 (copper resistivity 
). Find the current in the turn if the current in the solenoid winding is increased at a constant rate 
100 A/s. Neglect the magnetic field of the induction current.
Solution.
The magnetic field inside a long solenoid is uniform and equal
,
Where n number of turns per unit length, and I – instantaneous current value. Therefore, when choosing the direction of the normal to the coil surface along the field direction, the magnetic flux through this surface is equal to
,
Where 
- surface area of the coil.
As the current in the solenoid winding increases, the magnetic flux through the coil increases, and the resulting induced current is determined by the expression
,
Where 
, and the minus sign means that the induced current flows in the direction opposite to the direction of the positive circuit of the turn, consistent with the direction of the normal.
Then, the magnitude of the current through the turn at the moment of time t equal to
mA.
Example 11 .9.
A flat contour (Fig. 13), looking like two squares with sides a= 20 cm and b= 10 cm, is in a uniform magnetic field perpendicular to its plane. The field induction changes according to the law 
, Where B 0 = 10 mT and = 100 s –1. Find the amplitude of the induction current in the circuit if the resistance is unit of its length 
. Neglect the magnetic field of this current.
|
|
Solution.
The induced current in the frame is equal to
.
Figure 14 shows the direction of the magnetic field, as well as the normals to the surface of each of the squares that make up the contour, consistent with a single direction of positive detour. Taking this into account, the total magnetic flux through the circuit is equal to
|
|
.
Considering that the circuit resistance is equal to 
, let's find the amplitude of the induction current
on
Charge and magnetic flux change
Example 11.10.
Square made of resistance wire R= 1 Ohm, placed in a uniform magnetic field, induction vector 
which is perpendicular to the plane of the square. Square side length A= 1 cm. The magnitude of the magnetic field induction is initially equal to B=0.1 T and then it is reduced to zero. Find the value q charge, which as a result will move through the cross-section of the wire.
Solution.
The amount of electricity flowing through any cross-section of a circuit with resistance
R when the magnetic flux through the circuit changes by an amount 
, equals:
Note that the value q
does not depend on the nature of the time dependence of the change in magnetic flux, but is determined only by its initial and final values. Since the magnetic field induction varies from 
to zero, the increment of the magnetic flux penetrating the circuit is equal to
The amount of charge that flows through the wire is determined by the expression
Cl.
15.1 The phenomenon of electromagnetic induction.
15.1.1 Discovery of the phenomenon of electromagnetic induction by M. Faraday.
The discovery of the magnetic action of current by H. K. Oersted in 1820 proved that electrical and magnetic phenomena are interconnected. Theory A.M. Ampere reduced the numerous magnetic phenomena he studied to the interaction of electric currents, that is, moving electric charges. After the discovery of Oersted and the work of Ampere, the English scientist Michael Faraday came to the idea of the opposite process - the excitation of an electric current by magnetism: if an electric current generates a magnetic field, then why cannot a magnetic field excite an electric current? In 1822, an entry appeared in M. Faraday’s workbook in which the task was formulated: “Convert magnetism into electricity.” It took M. Faraday almost ten years of persistent and numerous experiments to solve the problem, which led to the discovery of the phenomenon of electromagnetic induction on August 29, 1831.
For a long time, M. Faraday carried a coil of wire and a permanent magnet in his pocket, in every free minute trying to come up with a new arrangement of wire and magnet that would lead to the appearance of an electric current. As has often happened in history, success came unexpectedly, although it took almost ten years to wait for it. To exclude the direct influence of the magnet on the device that records the current (galvanometer), M. Faraday placed magnets and conductors (usually coils) in one room, and the galvanometer in another. Having once again placed the coils and magnets, M. Faraday moved to another room to make sure once again that there was no electric current. Finally, one of the employees noticed that electric current arises only at the moment of relative movement of the conductor and magnet.
Now M. Faraday's experiments can be easily reproduced in a school laboratory. It is enough to connect a wire coil to a galvanometer and insert a permanent magnet inside the coil. When the magnet is moved into the coil, the galvanometer needle deflects, indicating the presence of current in the circuit (Fig. 104).
The current stops when the magnet is stationary. If you remove the magnet from the coil, then again the galvanometer registers the presence of current, only in the opposite direction. If you change the polarity of the magnet, the direction of the current also changes. The magnitude of the current depends on the speed of the magnet - the faster the magnet moves, the greater the strength of the resulting electric current. Similar results are obtained if the magnet is stationary and the coil moves.
In other words, the result depends only on the relative motion of the coil and magnet.
Further, M. Faraday showed that an electric current appears in the circuit even when it is in a time-varying magnetic field. To demonstrate this phenomenon, in previous experiments you can replace the permanent magnet with a coil connected to a direct current source (Fig. 105). The galvanometer registers current only at the moments when the current source is turned on and off. Please note that the coils are not connected to each other, the only connection between them is through a magnetic field.
Thus, in all cases, when the magnetic field changes, an electric current appears in a closed circuit, which indicates the appearance of an electromotive force in it. M. Faraday connected his reasoning about electromagnetic phenomena with the properties of lines of force, which he perceived as very real elastic threads and tubes. In such reasoning, an electric current occurs when the magnetic field lines move and cross a circuit, due to which an emf is induced in the circuit.
M. Faraday called the phenomenon of the occurrence of electric current in a circuit when the magnetic field changes phenomenon of electromagnetic induction.
Further, we will not strictly follow the reasoning and experiments of M. Faraday, because in his time the nature of electrical and magnetic phenomena was completely unknown: even electric current was not always associated with the movement of electric charges. Therefore, in our presentation we will use facts and ideas that became known much later.
15.1.2 Moving conductor in a magnetic field.
Today it is almost obvious that no configuration of a constant magnetic field can lead to the emergence of a direct electric current. To maintain current in an electrical circuit, as we know, there must be a source of external forces that does work to overcome resistance forces. The magnetic field acts only on moving charges, and the force acting on the charge (Lorentz force) is perpendicular to the velocity vector of the particle, so it does no work. Finally, if a stationary magnetic field could support an electric current, then this would be a direct path to the creation of a “perpetual motion machine,” that is, to “free” energy generation. Indeed, if the field is stationary, then its energy does not change, and a hypothetical electric current has energy and is capable of doing work. Therefore, for an EMF to occur in the circuit, there must be an external source of energy. Energy can enter the circuit due to the work of external forces.
Let's consider a group of simple thought experiments that allow theoretical description. Let a cylindrical conductor move in a constant magnetic field, so that the velocity vector \(~\vec \upsilon\) is perpendicular to the axis of the cylinder, and the magnetic field induction vector \(~\vec B\) is perpendicular to both the axis of the conductor and its speed ( Fig. 106). Together with the conductor, the free charges located inside it also move. From the side of the magnetic field, these charges will be acted upon by Lorentz forces, directed, in accordance with the left-hand rule, along the axis of the conductor.
The most well-known conductors are metals, where the free charges are negatively charged particles - electrons. However, here and further we will consider the movement of positively charged particles, because the direction of the current is taken to be the direction of the positive particles.
As a rule, free charges move in a conductor chaotically with equal probability in all directions, therefore in a stationary conductor the average value of the Lorentz force vector is zero. When the conductor moves, the chaotic thermal movement of free charges is superimposed by the directed movement of the conductor as a whole, due to which a non-zero resultant Lorentz force appears, which is the same for all particles. It is this constant force that leads to the emergence of electric current - the directed movement of charged particles. This gives good reason to ignore the violent but chaotic thermal movement.
Under the influence of the Lorentz force, free charges will begin to shift to the ends of the cylinder, where electric charges will be induced, described by surface densities ± σ . In turn, these charges will begin to create an electric field, the action of which on the charged particles will be directed in the direction opposite to the Lorentz force. At a constant speed of movement of the conductor, an equilibrium will be established in which the movement of charges will stop, but an electric field created by the induced charges will exist in the conductor. In the steady state, the Lorentz force \(F_L = q \upsilon B\) acting on the particle will be balanced by the force from the electric field \(F_(el) = q E\). Equating these forces, we determine the electric field strength in the conductor
\(~E = \upsilon B\) . (1)
Since the Lorentz force is the same at all points of the conductor, then the electric force must also be constant, that is, the resulting electric field is uniform. This electric field can also be characterized by the potential difference between the ends of the cylinder, which is equal to
\(~\Delta \varphi = E l = \upsilon B l\) , (2)
Where l- length of the conductor.
The Lorentz force acting on free charges in a conductor can be an external force, that is, it can lead to the emergence of an electric current in a closed circuit if it is connected to a moving conductor.
Let the conductor in question A.C. can slide on two parallel tires (rails) connected to each other (Fig. 107). The entire system is placed in a uniform magnetic field, the induction vector of which \(~\vec B\) is perpendicular to the plane of the buses. For simplicity, we will assume that the resistance of the tires and the moving conductor (jumper) is negligible compared to the resistance of the connecting resistor R. If an external force \(~\vec F\) is applied to a moving conductor, as shown in the figure, then it will begin to move. Under the influence of the Lorentz force, free charges in the conductor will begin to move, creating excess charges at the ends. These charges will create an electric field in the entire circuit formed by the jumper, bars and connecting resistor, so an electric current will arise in the circuit. The Lorentz force acting on the charges of a moving conductor will play the role of an external force, overcoming the forces acting from the electric field. The work done by this force to move a unit charge (that is, EMF) is equal to the product of the Lorentz force and the distance between the tires
\(~\varepsilon = \frac(1)(q) F_L l = \upsilon B l\) . (3)
Despite the fact that this expression for the emf completely coincides with formula (2) for the potential difference, its meaning is fundamentally different. Potential difference is the possible work of electric field forces; in the circuit under consideration, the direction of movement of charged particles is opposite to the direction of force from the electric field. The Lorentz force does work against the forces of the electric field, which is why it is extraneous. The electric field does positive work by “pushing” charged particles along the busbars and the connecting resistor (which in this case form an external circuit).
According to Ohm's law, the strength of the electric current generated in the circuit is equal to
\(~I = \frac(\varepsilon)(R) = \frac(\upsilon B l)(R)\) . (4)
Since an electric current flows through a conductor, an Ampere force equal to
\(~F_A = I B l = \frac(\upsilon B^2 l^2)(R)\) . (5)
The direction of this force is also determined by the “left hand rule”, with the help of which it is easy to determine that this force is directed in the direction opposite to the velocity vector, therefore formula (5) can be written in vector form
\(~\vec F_A = - \frac(B^2 l^2)(R) \vec \upsilon\) . (6)
By its nature, this force completely coincides with the force of viscous friction (proportional to the speed and directed in the opposite direction), therefore it is often called the force of magnetic viscosity.
Thus, in addition to the constant external force \(~\vec F\), the moving jumper is acted upon by the force of magnetic viscosity, which depends on the speed. The equation of Newton's second law for the jumper has the form (in projection onto the direction of the velocity vector):
\(~ma = F - \frac(B^2 l^2)(R) \upsilon\) . (7)
Under the influence of these forces, at first the jumper will move accelerated, and with increasing speed the acceleration modulus will decrease; finally, the jumper will move at a constant speed, which is called speed of steady motion\(~\overline (\upsilon)\). The value of this speed can be found from the condition \(F = F_A\), from which it follows
\(~\overline (\upsilon) = \frac(FR)(B^2 l^2)\) . (8)
Let us now consider the transformation of energy in this system in a steady state of motion. Over a period of time Δ t the jumper is displaced by a distance \(\Delta x = \overline (\upsilon) \Delta t\), therefore, the external force does work
\(~\Delta A = F \Delta x = F \overline (\upsilon) \Delta t = \frac(F^2 R)(B^2 l^2) \Delta t\) . (9)
During the same time, an amount of heat equal to
\(~Q = I^2 R \Delta t = \left (\frac(\upsilon B l)(R) \right)^2 R \Delta t = \frac(B^2 l^2)(R) \left (\frac(F R)(B^2 l^2) \right)^2 \Delta t = \frac(F^2 R)(B^2 l^2) \Delta t\) . (10)
As one would expect, the amount of heat released is exactly equal to the work done by the external force. Therefore, the source of energy for the electric current in the circuit is a device that moves the jumper (your hand could be such a device). If the action of this force stops, then the current in the circuit will disappear.
- Explain why, when the magnetic field induction tends to zero, the speed of the jumper, calculated by formula (8), tends to infinity.
- Explain why the speed of the jumper increases as the resistance of the resistor increases.
- Show that during the acceleration process, the work done by the external force is equal to the sum of the change in the kinetic energy of the jumper and the amount of heat released on the jumper.
In this case, the magnetic field plays the role of a kind of intermediary, facilitating the transformation of the energy of an external source (creating an external force) into the energy of an electric current, which is then converted into thermal energy. The external magnetic field itself does not change.
The clause about the external field in this case is not accidental; the electric current induced in the circuit creates its own magnetic field \(~\vec B"\). According to the gimlet rule, this field is directed opposite to the external field \(~\vec B\) (Fig. 108).
Let us now direct the direction of the external force to the opposite. In this case, the directions of movement of the jumper, the Lorentz force, the electric current in the circuit and the induction of the magnetic field of this current will change (Fig. 109). That is, in this case, the direction of the induction vector \(~\vec B"\) will coincide with the direction of the external field \(~\vec B\). Thus, the direction of the induced field is determined not only by the direction of the external field, but also by the direction of movement of the jumper .
We emphasize that the Ampere force, which plays the role of a viscous force, in this (and in all other) cases is opposite to the speed of movement of the jumper.
Let's try to formulate a general rule that allows us to determine the direction of the induction current. In Fig. 110 once again shows the diagrams of the experiments in question, if you look at them from above. Regardless of the direction of movement of the jumper, the induced emf in the circuit modulo is determined by formula (3), which we transform to the form
\(~\varepsilon = \upsilon B l = \frac(B l \Delta x)(\Delta t)\) , (11)
where Δ x = υ Δ t- the distance by which the jumper moves over a period of time Δ t. The expression in the numerator of this expression is equal to the change in magnetic flux through the circuit BlΔ x = Δ Φ , which occurred due to a change in its area. Now let's turn our attention to direction this EMF.
Of course, electromotive force, as the work of external forces, is a scalar quantity, so it is not entirely correct to talk about its direction.
However, in this case we are talking about the work of external forces along a contour, for which the positive direction of the bypass can be determined. To do this, you must first select the direction of the positive normal to the contour (obviously, the choice of this direction is arbitrary). As before, we will take the “counterclockwise” direction as positive when viewed from the end of the positive normal vector; accordingly, we will consider the “clockwise” direction to be negative (Fig. 111). In this sense, we can talk about the sign of the EMF: if, when going around in the positive direction (i.e., “counterclockwise”), external forces do positive work, then we will consider the value of the EMF to be positive and vice versa.
In this case, the positive direction of the normal is compatible with the direction of the external field induction vector. Obviously, the direction of the induced current coincides with the direction of the EMF.
According to the accepted definition, in the case A) the induced EMF and the current in the circuit are negative, in the case b) - are positive. We can generalize: the sign of the EMF is opposite to the sign of the change in magnetic flux through the circuit.
Thus, The induced emf in the circuit is equal to the change in magnetic flux through the circuit, taken with the opposite sign:
\(~\varepsilon = - \frac(\Delta \Phi)(\Delta t)\) . (12)
The resulting rule can be given a slightly different interpretation. Let us pay attention to the direction of the magnetic field created by the induction current: when the magnetic flux through the circuit increases, this field is opposite to the induction of the external field; when the magnetic flux decreases, the field of the induction current is directed in the same way as the external field. That is, induced current field in the circuit prevents change magnetic flux through this circuit. This rule is universal for this phenomenon and is called Lenz's rule .
This rule is closely related to the law of conservation of energy. Indeed, let us assume the opposite: let the direction of induction of the magnetic field created by the current in the circuit enhances change in magnetic flux through the circuit. In this case, we get a “self-accelerating” system: if the magnetic flux through the circuit accidentally increases, this will lead to the appearance of an electric current, which will further increase the flux through the circuit, which will lead to an even greater increase in the current, etc. Thus, it turns out that without an external source, the current in the circuit (and its energy) increases without limit, which contradicts the law of conservation of energy.
Please note that in this discussion we take into account the magnetic flux not only of the external field, but also the field created by the induced current. This field really needs to be taken into account: the Lorentz force acting on charged particles is determined by the total magnetic field at the location of the charge, regardless of the origin of this field. Thus, through a magnetic field, an electric current is able to influence itself - a changing current creates a changing magnetic field, which affects the electric current. This phenomenon is called self-induction, we will get to know him in more detail later. We note here that in many cases this phenomenon can be neglected, since the induced fields are usually quite weak.
It can also be shown that the direction of the magnetic viscosity force, which is always opposite to the speed of movement of the conductor in a magnetic field, is also associated with Lenz’s rule.
The broadest generalization of Lenz’s rule “for all occasions” sounds like this: the effect seeks to reduce the cause. Try to come up with examples yourself from various branches of science when this rule is true. It is more difficult (though not impossible) to come up with examples where this rule does not apply.
Let's consider another example of the occurrence of EMF in a conducting circuit moving in a magnetic field. Let the field be created by a cylindrical permanent magnet, and a circular loop L moves with speed \(~\vec \upsilon\) along the axis of this magnet, so that the plane of the contour remains perpendicular to the axis of the magnet all the time (Fig. 112).
In this case, the magnetic field is not uniform, but has axial symmetry. When a conductor moves in this field, the charged particles are acted upon by a Lorentz force directed along the conductor, which is constant in magnitude throughout the entire circuit. In this case, the Lorentz force again acts as an external force, leading to the emergence of an electric current in the circuit. The work done by this force to move a charge along a closed loop is non-zero, so this force is not potential. Let's calculate the induced emf arising in the circuit. A charged particle experiences a force equal to
\(~F = q \upsilon B_r\) , (13)
Where B r is the component of the induction vector perpendicular to the conductor velocity vector, in this case it is directed radially. Since this force throughout the entire contour is directed tangentially to the contour and is constant in magnitude, its work on moving a unit charge, that is, the emf, is equal to
\(~\varepsilon = \frac(1)(q) F_L = \upsilon B_r L\) , (14)
Where L- contour length. To find an expression for the radial component of the induction vector, we use the magnetic flux theorem. As a closed surface, we choose a thin cylinder of thickness Δ z = υ Δ t, the axis of which coincides with the axis of the magnet, and the radius is equal to the radius of the contour (Fig. 113).
Let us represent the magnetic flux through this surface as the sum of the fluxes through the lower base F 0, through the upper base F 1 and through the side surface
\(~\Phi_(bok) = B_r L \Delta z = B_r L \upsilon \Delta t\) . (15)
The sum of these flows is zero
\(~\Phi_0 + \Phi_1 + \Phi_(bok) = 0\) . (16)
Now let's correlate these surfaces with the contour under consideration.
The lateral surface of the cylinder is the surface that the contour in question sweeps, so we related the height of the cylinder to the speed of movement of the contour. The lower base rests on the position of the contour at some point in time t. By convention, the positive normal for a closed surface is the outer normal (shown in the figure). When describing the magnetic flux through the circuit, we agreed to consider the positive direction of the normal, the direction “along the field”. That is, the flow through the loop is opposite to the flow through part of the closed surface. Therefore in this case Φ 0 = −Φ (t), Where Φ (t) - flow through the circuit, at the moment of time t. The flow through the upper base is the flow through the circuit at the moment of time t + Δ t Φ 1 = Φ (t + Δ t). Another argument in favor of changing the sign in the flow through the lower base is that if we calculate the change in flow, then we must keep the direction of the normal unchanged.
Now we rewrite relation (16) in the form
\(~- \Phi(t) + \Phi(t + \Delta t) + B_r L \upsilon \Delta t = 0\) . (17)
From which we express the induced emf in the circuit (defined by formula (15))
\(~\varepsilon = B_r L \upsilon = - \frac(\Phi(t + \Delta t) - \Phi(t))(\Delta t) = -\frac(\Delta \Phi)(\Delta t )\) . (18)
We received the same formula for the induced emf in the circuit as in the previous example.
In the example considered, the magnetic flux through the circuit decreases, since as the distance from the magnet increases, the field induction decreases. Therefore, in accordance with the obtained formula and Lenz’s rule, the induced emf in the circuit is positive, in addition, the induced current creates a magnetic field directed in the same way as the field of a permanent magnet.
Please note that in the above derivation we did not make any assumptions about the dependence of the field induction vector on coordinates. The only assumption was about the axial symmetry of the field. However, it can also be removed; to do this, when calculating the EMF along a contour, it is simply necessary to divide the latter into small sections, and then sum up the work of the Lorentz force over all sections.
Assignments for independent work.
- Consider the direction of the field created by the induced current in the circuit in Fig. 112, show that Lenz's rule holds.
- Show that in the circuit shown in Fig. 112, the Ampere force acting on a circuit with an induced current is directed in the direction opposite to its speed.
- Let an arbitrary circuit shift from position 1 to position 2 in a short period of time in an arbitrary constant magnetic field. Using the expression for the Lorentz force and the magnetic flux theorem, prove in the general case formula (18) for the induced emf in the circuit (Fig. 114).