9.5. Induction current
9.5.1. Thermal action induction current
The occurrence of EMF leads to the appearance in the conducting circuit induction current, the strength of which is determined by the formula
i i = | ℰ i | R,
where ℰ i is the induction emf that occurs in the circuit; R is the loop resistance.
When an induction current flows in the circuit, heat is released, the amount of which is determined by one of the expressions:
Q i = I i 2 R t , Q i = ℰ i 2 t R , Q i = I i | ℰ i | t ,
where I i - the strength of the induction current in the circuit; R is the loop resistance; t - time; ℰ i - EMF of induction that occurs in the circuit.
Induction current power calculated by one of the formulas:
P i = I i 2 R , P i = ℰ i 2 R , P i = I i | ℰ i | ,
where I i - the strength of the induction current in the circuit; R is the loop resistance; ℰ i - EMF of induction that occurs in the circuit.
When an inductive current flows in a conducting circuit, a charge is transferred through the cross-sectional area of the conductor, the value of which is calculated by the formula
q i = I i ∆t ,
where I i - the strength of the induction current in the circuit; Δt is the time interval during which the inductive current flows through the circuit.
Example 21. A ring made of wire with a resistivity of 50.0 ⋅ 10 −10 Ohm ⋅ m is placed in a uniform magnetic field with an induction of 250 mT. The length of the wire is 1.57 m, and its cross-sectional area is 0.100 mm 2 . What is the maximum charge that will pass through the ring when the field is turned off?
Solution . The appearance of the induction EMF in the ring is caused by a change in the flux of the induction vector penetrating the plane of the ring when the magnetic field is turned off.
The magnetic field induction flux through the ring area is determined by the formulas:
- before turning off the magnetic field
Ф 1 = B 1 S cos α,
where B 1 is the initial value of the magnetic field induction module, B 1 = 250 mT; S is the area of the ring; α is the angle between the directions of the magnetic induction vector and the vector of the normal (perpendicular) to the plane of the ring;
- after turning off the magnetic field
Ф 2 = B 2 S cos α = 0,
where B 2 is the value of the induction modulus after turning off the magnetic field, B 2 = 0.
∆Ф = Ф 2 − Ф 1 = −Ф 1,
or, taking into account the explicit form Ф 1 ,
∆Ф = −B 1 S cos α.
The average value of the EMF of induction that occurs in the ring when the field is turned off,
| ℰ i | = | Δ Ф Δ t | = | − B 1 S cos α Δ t | = B 1 S | cosα | Δt,
where ∆t is the time interval during which the field is turned off.
The presence of the induction EMF leads to the appearance of an inductive current; the strength of the induction current is determined by Ohm's law:
i i = | ℰ i | R = B 1 S | cosα | R ∆ t ,
where R is the resistance of the ring.
When an inductive current flows through the ring, an inductive charge is transferred
q i = I i Δ t = B 1 S | cosα | R.
The maximum value of the charge corresponds to the maximum value of the cosine function (cos α = 1):
q i max \u003d I i Δ t \u003d B 1 S R .
The resulting formula determines the maximum value of the charge that will pass through the ring when the field is turned off.
However, to calculate the charge, it is necessary to obtain expressions that will allow you to find the area of \u200b\u200bthe ring and its resistance.
The area of the ring is the area of a circle with radius r, the perimeter of which is determined by the circumference formula and coincides with the length of the wire from which the ring is made:
l = 2πr ,
where l is the length of the wire, l = 1.57 m.
It follows that the radius of the ring is determined by the relation
r \u003d l 2 π,
and its area is
S \u003d π r 2 \u003d π l 2 4 π 2 \u003d l 2 4 π.
Ring resistance is given by the formula
R = ρ l S 0 ,
where ρ is the resistivity of the wire material, ρ = 50.0 × 10 −10 Ohm ⋅ m; S 0 - cross-sectional area of the wire, S 0 = = 0.100 mm 2.
Let us substitute the obtained expressions for the area of the ring and its resistance into the formula that determines the desired charge:
q i max = B 1 l 2 S 0 4 π ρ l = B 1 l S 0 4 π ρ .
Let's calculate:
q i max = 250 ⋅ 10 − 3 ⋅ 1.57 ⋅ 0.100 ⋅ 10 − 6 4 ⋅ 3.14 ⋅ 50.0 ⋅ 10 − 10 = 0.625 C = 625 mC.
When the field is turned off, a charge equal to 625 mC passes through the ring.
Example 22. A circuit with an area of 2.0 m 2 and a resistance of 15 mΩ is in a uniform magnetic field, the induction of which increases by 0.30 mT per second. Find the maximum possible power of the induction current in the circuit.
Solution . The appearance of the induction EMF in the circuit is caused by a change in the flux of the induction vector penetrating the plane of the circuit, with a change in the magnetic field induction over time.
The change in the flux of the magnetic field induction vector is determined by the difference
∆Ф = ∆BS cos α,
where ∆B is the change in the magnetic field induction modulus for the selected time interval; S - area bounded by the contour, S = 2.0 m 2; α is the angle between the directions of the magnetic induction vector and the normal vector (perpendicular) to the contour plane.
The average value of the EMF of the induction that occurs in the circuit when the magnetic field induction changes:
| ℰ i | = | Δ Ф Δ t | = | Δ B S cos α Δ t | = ∆BS | cosα | Δt,
where ∆B /∆t is the rate of change of the modulus of the magnetic field induction vector over time, ∆B /∆t = 0.30 mT/s.
The appearance of the EMF of induction leads to the appearance of an inductive current; the strength of the induction current is determined by Ohm's law:
i i = | ℰ i | R = ∆BS | cosα | R ∆ t ,
where R is the loop resistance.
Induction current power
P i = I i 2 R = (Δ B Δ t) 2 S 2 R cos 2 α R 2 = (Δ B Δ t) 2 S 2 cos 2 α R .
The maximum value of the induction current power corresponds to the maximum value of the cosine function (cos α = 1):
P i max \u003d (Δ B Δ t) 2 S 2 R.
Let's calculate:
P i max \u003d (0.30 ⋅ 10 - 3) 2 (2.0) 2 15 ⋅ 10 - 3 \u003d 24 ⋅ 10 - 6 W \u003d 24 μW.
The maximum power of the induction current in this circuit is 24 μW.
The relationship between electric and magnetic fields has been noticed for a very long time. This connection was discovered in the 19th century by the English physicist Faraday and gave it a name. It appears at the moment when the magnetic flux penetrates the surface of a closed circuit. After a change in the magnetic flux occurs for a certain time, an electric current appears in this circuit.
The relationship of electromagnetic induction and magnetic flux
The essence of the magnetic flux is displayed by the well-known formula: Ф = BS cos α. In it, F is a magnetic flux, S is the surface of the contour (area), B is the vector of magnetic induction. The angle α is formed due to the direction of the magnetic induction vector and the normal to the contour surface. It follows that the magnetic flux will reach the maximum threshold at cos α = 1, and the minimum threshold at cos α = 0.
In the second variant, the vector B will be perpendicular to the normal. It turns out that the flow lines do not cross the contour, but only slide along its plane. Therefore, the characteristics will be determined by the lines of the vector B that intersect the surface of the contour. For calculation, Weber is used as a unit of measurement: 1 wb \u003d 1v x 1s (volt-second). Another, smaller unit of measure is the maxwell (µs). It is: 1 wb \u003d 108 μs, that is, 1 μs \u003d 10-8 wb.
For research by Faraday, two wire spirals were used, isolated from each other and placed on a wooden coil. One of them was connected to an energy source, and the other to a galvanometer designed to record small currents. At that moment, when the circuit of the original spiral closed and opened, in the other circuit the arrow of the measuring device deviated.
Conducting research on the phenomenon of induction
In the first series of experiments, Michael Faraday inserted a magnetized metal bar into a coil connected to a current, and then pulled it out (Fig. 1, 2).
1 
2 
When a magnet is placed in a coil connected to a measuring device, an inductive current begins to flow in the circuit. If the magnetic bar is removed from the coil, the induction current still appears, but its direction is already reversed. Consequently, the parameters of the induction current will be changed in the direction of the bar and depending on the pole with which it is placed in the coil. The strength of the current is affected by the speed of movement of the magnet.
In the second series of experiments, a phenomenon is confirmed in which a changing current in one coil causes an induction current in another coil (Fig. 3, 4, 5). This happens at the moments of closing and opening the circuit. The direction of the current will depend on whether the electrical circuit closes or opens. In addition, these actions are nothing more than ways to change the magnetic flux. When the circuit is closed, it will increase, and when it is opened, it will decrease, simultaneously penetrating the first coil.
3 
4 
5 
As a result of the experiments, it was found that the occurrence of an electric current inside a closed conducting circuit is possible only when they are placed in an alternating magnetic field. At the same time, the flow can change in time by any means.
The electric current that appears under the influence of electromagnetic induction is called induction, although this will not be a current in the conventional sense. When a closed circuit is in a magnetic field, an EMF is generated with an exact value, and not a current depending on different resistances.
This phenomenon is called the EMF of induction, which is reflected by the formula: Eind = - ∆F / ∆t. Its value coincides with the rate of change of the magnetic flux penetrating the surface of a closed loop, taken with a negative value. The minus present in this expression is a reflection of Lenz's rule.
Lenz's rule for magnetic flux
A well-known rule was derived after a series of studies in the 30s of the 19th century. It is formulated in the following way:
The direction of the induction current, excited in a closed circuit by a changing magnetic flux, affects the magnetic field it creates in such a way that it, in turn, creates an obstacle to the magnetic flux that causes the appearance of the inductive current.
When the magnetic flux increases, that is, it becomes Ф > 0, and the induction EMF decreases and becomes Eind< 0, в результате этого появляется электроток с такой направленностью, при которой под влиянием его магнитного поля происходит изменение потока в сторону уменьшения при его прохождении через плоскость замкнутого контура.
If the flow decreases, then the reverse process occurs when F< 0 и Еинд >0, that is, the action of the magnetic field of the induction current, there is an increase in the magnetic flux passing through the circuit.
The physical meaning of Lenz's rule is to reflect the law of conservation of energy, when when one quantity decreases, the other increases, and, conversely, when one quantity increases, the other will decrease. Various factors also affect the induction emf. When a strong and weak magnet is alternately inserted into the coil, the device will respectively show a higher value in the first case, and a lower value in the second. The same thing happens when the speed of the magnet changes.
The figure below shows how the direction of the induction current is determined using the Lenz rule. The blue color corresponds to the lines of force of the magnetic fields of the induction current and the permanent magnet. They are located in the direction of the north-south poles that are present in every magnet.
The changing magnetic flux leads to the emergence of an inductive electric current, the direction of which causes opposition from its magnetic field, which prevents changes in the magnetic flux. In this regard, the lines of force of the magnetic field of the coil are directed in the direction opposite to the lines of force of the permanent magnet, since its movement occurs in the direction of this coil.
To determine the direction of the current, it is used with a right-hand thread. It must be screwed in in such a way that the direction of its forward movement coincides with the direction of the induction lines of the coil. In this case, the directions of the induction current and the rotation of the gimlet handle will coincide.
The figure shows the direction of the inductive current that occurs in a short-circuited wire coil when the coil is moved relative to it.
magnet. Mark which of the following statements are correct and which are incorrect.
A. The magnet and the coil are attracted to each other.
B. Inside the coil, the magnetic field of the induction current is directed upwards.
B. Inside the coil, the lines of magnetic induction of the field of the magnet are directed upwards.
D. The magnet is removed from the coil.
2. What frames of reference are inertial and non-inertial? Give examples.
3. What is the property of bodies called inertia? What is the value of inertia?
4. What is the relationship between the masses of bodies and the modules of accelerations that they receive during interaction?
5. What is strength and how is it characterized?
6. Statement of Newton's 2nd law? What is its mathematical notation?
7. How is Newton's 2nd law formulated in impulsive form? His math notation?
8. What is 1 Newton?
9. How does a body move if a force is applied to it that is constant in magnitude and direction? What is the direction of the acceleration caused by the force acting on it?
10. How is the resultant of forces determined?
11. How is Newton's 3rd law formulated and written down?
12. How are the accelerations of interacting bodies directed?
13. Give examples of the manifestation of Newton's 3rd law.
14. What are the limits of applicability of all Newton's laws?
15. Why can we consider the Earth as an inertial frame of reference if it moves with centripetal acceleration?
16. What is deformation, what types of deformation do you know?
17. What force is called the force of elasticity? What is the nature of this force?
18. What are the features of the elastic force?
19. How is the elastic force directed (support reaction force, thread tension force?)
20. How is Hooke's law formulated and written? What are its limits of applicability? Plot a graph illustrating Hooke's law.
21. How is the law of universal gravitation formulated and written down, when is it applicable?
22. Describe the experiments to determine the value of the gravitational constant?
23. What is the gravitational constant, what is its physical meaning?
24. Does the work of the gravitational force depend on the shape of the trajectory? What is the work done by gravity in a closed loop?
25. Does the work of the elastic force depend on the shape of the trajectory?
26. What do you know about gravity?
27. How is free fall acceleration calculated on Earth and other planets?
28. What is the first cosmic speed? How is it calculated?
29. What is called free fall? Does the acceleration of free fall depend on the mass of the body?
30. Describe the experience of Galileo Galilei, proving that all bodies in a vacuum fall with the same acceleration.
31. What force is called the force of friction? Types of friction forces?
32. How is the force of sliding and rolling friction calculated?
33. When does the static friction force arise? What is it equal to?
34. Does the force of sliding friction depend on the area of the contact surfaces?
35. On what parameters does the force of sliding friction depend?
36. What determines the force of resistance to the movement of a body in liquids and gases?
37. What is called body weight? What is the difference between the weight of a body and the force of gravity acting on a body?
38. In what case is the weight of the body numerically equal to the modulus of gravity?
39. What is weightlessness? What is overload?
40. How to calculate the weight of a body during its accelerated movement? Does the weight of a body change if it moves along a fixed horizontal plane with acceleration?
41. How does the weight of a body change when it moves along the convex and concave parts of the circle?
42. What is the algorithm for solving problems when a body moves under the action of several forces?
43. What force is called the Archimedes Force or the buoyant force? On what parameters does this force depend?
44. What formulas can be used to calculate the force of Archimedes?
45. Under what conditions does a body in a liquid float, sink, float?
46. How does the depth of immersion in a liquid of a floating body depend on its density?
47. Why are balloons filled with hydrogen, helium or hot air?
48. Explain the influence of the rotation of the Earth around its axis on the value of the acceleration of free fall.
49. How does the value of gravity change when: a) the removal of the body from the surface of the Earth, B) when the body moves along the meridian, parallel
electrical circuit?
3. What is the physical meaning of EMF? Define volt.
4. Connect the voltmeter for a short time to a source of electrical energy, observing the polarity. Compare his readings with the calculation based on the results of the experiment.
5. What determines the voltage at the terminals of current sources?
6. Using the measurement results, determine the voltage on the external circuit (if the work was done by method I), the resistance of the external circuit (if the work was done by method II).
6 question in nesting calculation
Help me please!1. Under what conditions do friction forces appear?
2. What determines the modulus and direction of the static friction force?
3. Within what limits can the static friction force change?
4. What force imparts acceleration to a car or locomotive?
5. Can the force of sliding friction increase the speed of a body?
6. What is the main difference between the resistance force in liquids and gases and the friction force between two solid bodies?
7. Give examples of the beneficial and harmful effects of friction forces of all types
As we have already found out, electric current is capable of generating magnetic fields. The question arises: can a magnetic field cause the appearance of an electric current? This problem was solved by the English physicist Michael Faraday, who discovered the phenomenon of electromagnetic induction in 1831. A coiled conductor closes on a galvanometer (Fig. 3.19). If a permanent magnet is pushed into the coil, the galvanometer will show the presence of current for the entire time period while the magnet is moving relative to the coil. When the magnet is pulled out of the coil, the galvanometer shows the presence of a current in the opposite direction. A change in the direction of the current occurs when the retractable or retractable pole of the magnet changes.
Similar results were observed when replacing a permanent magnet with an electromagnet (coil with current). If both coils are fixed motionless, but the current value is changed in one of them, then at this moment an induction current is observed in the other coil.
THE PHENOMENON OF ELECTROMAGNETIC INDUCTION consists in the occurrence of an electromotive force (emf) of induction in a conducting circuit, through which the flux of the magnetic induction vector changes. If the circuit is closed, then an induction current arises in it.
Discovery of the phenomenon of electromagnetic induction:
1) showed relationship between electric and magnetic field;
2) suggested method of generating electric current using a magnetic field.
Main properties of induction current:
1. Induction current always occurs when there is a change in the flux of magnetic induction coupled to the circuit.
2. The strength of the induction current does not depend on the method of changing the flux of magnetic induction, but is determined only by the rate of its change.
Faraday's experiments found that the magnitude of the electromotive force of induction is proportional to the rate of change of the magnetic flux penetrating the conductor circuit (Faraday's law of electromagnetic induction)
Or , (3.46)
where (dF) is the change in flux over time (dt). MAGNETIC FLUX or FLOW OF MAGNETIC INDUCTION is called the value, which is determined on the basis of the following relation: ( magnetic flux through a surface area S): Ф=ВScosα, (3.45), angle a is the angle between the normal to the surface under consideration and the direction of the magnetic field induction vector
unit of magnetic flux in the SI system is called weber- [Wb \u003d Tl × m 2].
The sign "-" in the formula means that the emf. induction causes an induction current, the magnetic field of which counteracts any change in the magnetic flux, i.e. at >0 e.m.f. induction e AND<0 и наоборот.
emf induction is measured in volts
To find the direction of the induction current, there is Lenz's rule (the rule was established in 1833): the induction current has such a direction that the magnetic field it creates tends to compensate for the change in the magnetic flux that caused this induction current.
For example, if you push the north pole of the magnet into the coil, that is, increase the magnetic flux through its turns, an induction current arises in the coil in such a direction that a north pole appears at the end of the coil closest to the magnet (Fig. 3.20). So, the magnetic field of the induction current tends to neutralize the change in the magnetic flux that caused it.
Not only an alternating magnetic field generates an induction current in a closed conductor, but also when a closed conductor of length l moves in a constant magnetic field (B) at a speed v, an emf arises in the conductor:
a (B Ùv) (3.47)
As you already know, electromotive force in the chain is the result of external forces. When the conductor moves in a magnetic field, the role of external forces performs Lorentz force(which acts from the side of the magnetic field on a moving electric charge). Under the action of this force, a separation of charges occurs and a potential difference arises at the ends of the conductor. emf induction in a conductor is the work of moving unit charges along the conductor.
Direction of induction current can be defined according to the right hand rule:Vector B enters the palm, the abducted thumb coincides with the direction of the conductor's velocity, and 4 fingers indicate the direction of the induction current.
Thus, an alternating magnetic field causes the appearance of an induced electric field. It not potentially(as opposed to electrostatic), because Work by the displacement of a single positive charge equal to emf. induction, not zero.
Such fields are called vortex. The lines of force of the vortex electric field - locked in on themselves in contrast to the lines of electrostatic field strength.
emf induction occurs not only in neighboring conductors, but also in the conductor itself when the magnetic field of the current flowing through the conductor changes. Emf occurrence. in any conductor, when the current strength changes in it (hence, the magnetic flux in the conductor) is called self-induction, and the current induced in this conductor is self-induction current.
The current in a closed circuit creates a magnetic field in the surrounding space, the strength of which is proportional to the strength of the current I. Therefore, the magnetic flux Ф penetrating the circuit is proportional to the strength of the current in the circuit
Ф=L×I, (3.48).
L is the coefficient of proportionality, which is called the coefficient of self-induction, or, simply, inductance. The inductance depends on the size and shape of the circuit, as well as on the magnetic permeability of the medium surrounding the circuit.
In this sense, the inductance of the circuit - analogue the electric capacitance of a solitary conductor, which also depends only on the shape of the conductor, its dimensions and the permittivity of the medium.
The unit of inductance is henry (H): 1H - the inductance of such a circuit, the magnetic flux of self-induction of which at a current of 1A is 1Wb (1Hn \u003d 1Wb / A \u003d 1V s / A).
If L=const, then emf. self-induction can be represented in the following form:
, or 
, (3.49)
where DI (dI) is the change in current in the circuit containing the inductor (or circuit) L, during the time Dt (dt). The sign "-" in this expression means that the emf. self-induction prevents a change in current (i.e., if the current in a closed circuit decreases, then the emf of self-induction leads to a current in the same direction and vice versa).
One of the manifestations of electromagnetic induction is the occurrence of closed induction currents in continuous conductive media: metallic bodies, electrolyte solutions, biological organs, etc. Such currents are called eddy currents or Foucault currents. These currents arise when a conducting body moves in a magnetic field and/or when the induction of the field in which the bodies are placed changes with time. The strength of the Foucault currents depends on the electrical resistance of the bodies, as well as on the rate of change of the magnetic field.
Foucault currents also obey Lenz's rule : their magnetic field is directed so as to counteract the change in magnetic flux that induces eddy currents.
Therefore, massive conductors are decelerated in a magnetic field. In electrical machines, in order to minimize the effect of Foucault currents, the cores of transformers and the magnetic circuits of electrical machines are assembled from thin plates isolated from each other by a special varnish or scale.
Eddy currents cause strong heating of conductors. Joule heat generated by Foucault currents, used in induction metallurgical furnaces for melting metals, according to the Joule-Lenz law.
