Derivation of the formula for the emf of induction in moving conductors. Emf of induction in moving conductors. EMF of induction in a flat coil rotating in a magnetic field
A rectilinear conductor AB moves in a magnetic field with induction B along conductive tires that are closed to a galvanometer.
Electric charges moving with a conductor in a magnetic field are affected by the Lorentz force:
Fl \u003d / q / vB sin a
Its direction can be determined by the rule of the left hand.
Under the action of the Lorentz force inside the conductor, positive and negative charges are distributed along the entire length of the conductor l
The Lorentz force is in this case a third-party force, and an EMF of induction occurs in the conductor, and a potential difference arises at the ends of the conductor AB.
The reason for the induction EMF in a moving conductor is explained by the action of the Lorentz force on free charges.
Getting ready for the test!
1. In what direction of movement of the circuit in a magnetic field will an induction current occur in the circuit?
2. Indicate the direction of the induction current in the circuit when it is introduced into a uniform magnetic field.
3. How will the magnetic flux in the frame change if the frame is rotated 90 degrees from position 1 to position 2?
4. Will there be an induction current in the conductors if they move as shown in the figure?
5. Determine the direction of the induction current in the AB conductor moving in a uniform magnetic field.
6. Indicate the correct direction of the induction current in the circuits.
Electromagnetic field - Cool physics
Or, conversely, a moving magnetic field crosses a fixed conductor; or when the conductor and the magnetic field, moving in space, move one relative to the other;
Thus, any change in time of the value penetrating a closed loop (coil, frame) is accompanied by the appearance of an induced emf in the conductor.
A = U × I × t = I² × r × t(J) .
The power consumed will be equal to:
P email = U × I = I² × r(W) ,
where we determine the current in the circuit:
| (1) |
However, we know that a current-carrying conductor placed in a magnetic field will experience a force from the field tending to move in the direction determined by the left hand rule. During its movement, the conductor will cross the magnetic field lines of the field and, according to the law of electromagnetic induction, an induced emf will appear in it. The direction of this EMF, determined by the right hand rule, will be the reverse of the current I. Let's call it the back EMF E arr. Value E arr according to the law of electromagnetic induction will be equal to:
E arr = B × l × v(AT) .
For a closed circuit we have:
U - E arr = I × r
| U = E arr + I × r , | (2) |
where is the current in the circuit
| (3) |
Comparing expressions (1) and (3), we see that in a conductor moving in a magnetic field, for the same values U and r the current will be less than with a fixed conductor.
Multiplying the resulting expression (2) by I, we get:
U × I = E arr × I + I² × r .
As E arr = B × l × v, then
U × I = B × l × v × I + I² × r .
Given that B × l × I = F and F × v = P fur, we have:
U × I = F × v + I² × r
P = P fur + P Em.
The last expression shows that when a current-carrying conductor moves in a magnetic field, the power of the voltage source is converted into thermal and mechanical powers.
Occurrence in the conductor of EMF induction
If placed in conductor and move it so that during its movement it crosses the field lines of force, then a, called the EMF of induction.
EMF of induction will occur in the conductor even if the conductor itself remains motionless, and the magnetic field moves, crossing the conductor with its lines of force.
If the conductor in which the induction EMF is induced is closed to any external circuit, then under the action of this EMF, a current will flow through the circuit, called induction current.
EMF induction phenomenon in a conductor when it is crossed by magnetic field lines is called electromagnetic induction.
Electromagnetic induction is the reverse process, i.e., the conversion of mechanical energy into electrical energy.
The phenomenon of electromagnetic induction has found the widest application in. The device of various electrical machines is based on its use.
The magnitude and direction of the induction emf
Let us now consider what will be the magnitude and direction of the EMF induced in the conductor.
The value of the EMF of induction depends on the number of field lines of force crossing the conductor per unit time, i.e., on the speed of the conductor in the field.
The magnitude of the induced emf is directly dependent on the speed of the conductor in a magnetic field.
The magnitude of the induced emf also depends on the length of that part of the conductor that is intersected by the field lines. The greater part of the conductor is crossed by the field lines, the greater the EMF is induced in the conductor. And, finally, the stronger the magnetic field, i.e., the greater its induction, the greater the EMF occurs in the conductor crossing this field.
So, the magnitude of the EMF of induction that occurs in the conductor when it moves in a magnetic field is directly proportional to the induction of the magnetic field, the length of the conductor and the speed of its movement.
This dependence is expressed by the formula E = Blv,
where E is the induction emf; B - magnetic induction; I - conductor length; v - the speed of the conductor.
It must be firmly remembered that in a conductor moving in a magnetic field, an EMF of induction occurs only if this conductor is crossed by magnetic field lines. If the conductor moves along the field lines of force, i.e., does not cross, but, as it were, slides along them, then no EMF is induced in it. Therefore, the above formula is valid only when the conductor moves perpendicular to the magnetic field lines.
The direction of the induced emf (as well as the current in the conductor) depends on which direction the conductor is moving. To determine the direction of the induced emf, there is a right hand rule.
If you hold the palm of your right hand so that the magnetic field lines enter it, and the bent thumb indicates the direction of movement of the conductor, then the extended four fingers indicate the direction of the induced EMF and the direction of the current in the conductor.
Right hand rule
EMF of induction in the coil
We have already said that in order to create an EMF induction in a conductor, it is necessary to move either the conductor itself or the magnetic field in a magnetic field. In both cases, the conductor must be crossed by magnetic field lines, otherwise the EMF will not be induced. The induced EMF, and hence the induced current, can be obtained not only in a straight conductor, but also in a conductor wound into a coil.
When moving inside a permanent magnet, an EMF is induced in it due to the fact that the magnetic flux of the magnet crosses the turns of the coil, i.e., in exactly the same way as it was when a straight conductor moved in the field of a magnet.
If the magnet is lowered into the coil slowly, then the emf that arises in it will be so small that the arrow of the device may not even deviate. If, on the contrary, the magnet is quickly introduced into the coil, then the deflection of the arrow will be large. This means that the magnitude of the induced EMF, and hence the current strength in the coil, depends on the speed of the magnet, that is, on how quickly the field lines cross the turns of the coil. If we now alternately introduce a strong magnet into the coil at the same speed, and then a weak one, then we can see that with a strong magnet, the arrow of the device will deviate by a larger angle. Means, the magnitude of the induced emf, and hence the current strength in the coil, depends on the magnitude of the magnetic flux of the magnet.
And, finally, if the same magnet is introduced at the same speed, first into a coil with a large number of turns, and then with a much smaller number, then in the first case the arrow of the device will deviate by a larger angle than in the second. This means that the magnitude of the induced EMF, and hence the current strength in the coil, depends on the number of its turns. The same results can be obtained if an electromagnet is used instead of a permanent magnet.
The direction of the EMF of induction in the coil depends on the direction of movement of the magnet. How to determine the direction of the EMF of induction, says the law established by E. X. Lenz.
Lenz's law for electromagnetic induction
Any change in the magnetic flux inside the coil is accompanied by the appearance of an induction EMF in it, and the faster the magnetic flux penetrating the coil changes, the greater the EMF is induced in it.
If the coil in which the induction EMF is created is closed to an external circuit, then an induction current flows through its turns, creating a magnetic field around the conductor, due to which the coil turns into a solenoid. It turns out in such a way that a changing external magnetic field causes an induction current in the coil, which, in turn, creates its own magnetic field around the coil - the current field.
Studying this phenomenon, E. X. Lenz established a law that determines the direction of the induction current in the coil, and, consequently, the direction of the induction EMF. The induction emf that occurs in the coil when the magnetic flux changes in it creates a current in the coil in such a direction that the magnetic flux of the coil created by this current prevents a change in the extraneous magnetic flux.
Lenz's law is valid for all cases of current induction in conductors, regardless of the shape of the conductors and on how the change in the external magnetic field is achieved.
When a permanent magnet moves relative to a wire coil attached to the terminals of a galvanometer, or when the coil moves relative to a magnet, an induction current occurs.
Induction currents in massive conductors
A changing magnetic flux is capable of inducing an EMF not only in coil turns, but also in massive metal conductors. Penetrating the thickness of a massive conductor, the magnetic flux induces an EMF in it, which creates induction currents. These so-called ones propagate along the massive conductor and are short-circuited in it.
The cores of transformers, the magnetic cores of various electrical machines and apparatuses are just those massive conductors that are heated by the induction currents that arise in them. This phenomenon is undesirable, therefore, in order to reduce the magnitude of induction currents, parts of electrical machines and transformer cores are made not massive, but consisting of thin sheets isolated from one another by paper or a layer of insulating varnish. Due to this, the path of propagation of eddy currents along the mass of the conductor is blocked.
But sometimes in practice eddy currents are also used as useful currents. The use of these currents is based, for example, on the operation of the so-called magnetic dampers of the moving parts of electrical measuring instruments.
When a rectilinear conductor moves in a magnetic field, e occurs at the ends of the conductor. d.s. induction. It can be calculated not only by the formula, but also by the formula e. d.s.
induction in a straight conductor. It comes out like this. Equate formulas (1) and (2) § 97:
BIls = EIΔt, from here
where s/Δt=v is the speed of the conductor. Therefore e. d.s. induction when the conductor moves perpendicular to the magnetic field lines
E=Blv.
If the conductor moves at a speed v (Fig. 148, a), directed at an angle α to the induction lines, then the speed v is decomposed into components v 1 and v 2. The component is directed along the lines of induction and does not cause e in it when the conductor moves. d.s. induction. In the conductor e. d.s. is induced only by the component v 2 \u003d v sin α directed perpendicular to the lines of induction. In this case e. d.s. induction will
E \u003d Blv sin α.
This is the formula e. d.s. induction in a straight conductor.
So, when a straight conductor moves in a magnetic field, e is induced in it. d.s., the value of which is directly proportional to the active length of the conductor and the normal component of the speed of its movement.
If instead of one straight conductor we take a frame, then when it rotates in a uniform magnetic field, e. d.s. in two of its sides (see Fig. 138). In this case e. d.s. induction will E \u003d 2 Blv sin α. Here l is the length of one active side of the frame. If the latter consists of n turns, then e appears in it. d.s. induction
E = 2nBlv sin α.
That e. d.s. induction depends on the speed v of rotation of the frame and on the induction B of the magnetic field, can be seen in such an experiment (Fig. 148, b). When the armature of the current generator rotates slowly, the lamp burns dimly: a small e. d.s. induction. With an increase in the speed of rotation of the armature, the lamp burns brighter: a large e. d.s. induction. At the same armature rotation speed, we remove one of the magnets, thereby reducing the magnetic field induction. The lamp is dimly lit: e. d.s. induction has decreased.
Task 35. Straight conductor length 0.6 m flexible conductors attached to a current source, e. d.s. whom 24 in and internal resistance 0.5 ohm. The conductor is in a uniform magnetic field with induction 0.8 tl, the lines of induction of which are directed towards the reader (Fig. 149). The resistance of the entire external circuit 2.5 ohm. Determine the strength of the current in the conductor if it moves perpendicular to the lines of induction with a speed 10 m/s What is the current strength in a fixed conductor?
A metal conductor contains a large number of free electrons that move randomly. If you move a conductor in a magnetic field perpendicular to the lines of force, then the field will deflect the electrons moving along with the conductor, and they will begin to move, that is, there will be electromotive force (EMF). It is called electromagnetic induction(induce - induce).
Under the action of the EMF, the electrons will move and accumulate at one end of the conductor, and at the other there will be a lack of electrons, that is, a positive charge will arise potential difference, or electrical voltage.
If you connect such a conductor to an external circuit (close the path), then under the influence of the potential difference, a current will flow.
If the conductor is moved along the lines of force, then the field will not act on the charges, the EMF, the voltage will not arise, the current will not flow.
This EMF is called EMF induction. It is determined by Faraday's law:
· EMF induction is equal to the product of the speed of the conductor V, magnetic induction AT and active conductor length L
Its direction is determined by right hand rule:
· 
If the right hand is placed in a magnetic field so that the lines of force will enter the palm, and the bent thumb will show the direction of movement of the conductor, then four extended fingers will show the direction of the EMF.
EMF will be induced at any intersection of the conductor and the magnetic field. That is, you can move the conductor, you can field, and you can change the magnetic field.
Then the EMF is determined according to Maxwell:
The emf induced in the circuit as a result of its crossing by a changing magnetic flux is equal to the rate of change of this flux.
e= - ΔF/Δt
Where ΔF \u003d F 1 - F 2 change in magnetic flux, Wb
Δt is the time during which the magnetic flux changed, sec.
Lenz's rule: The induced emf is in such a direction that the current it creates opposes the change in magnetic flux.
EMF of self-induction.
If the current in the conductor changes, the magnetic flux created by it also changes. Propagating in space, this magnetic flux crosses not only neighboring conductors, but also its own, which means that an EMF is induced in its own conductor. It is called EMF self-induction.
EMF self-induction- this is the EMF that occurs in the conductor, with a change in its own current and magnetic flux.
It occurs with every change in current and is directed so as not to allow it to change. When the current decreases, it is directed along with it and supports the current; when the current increases, it is directed against and weakens it.
The ability of a conductor (coil) to create an EMF of self-induction is called inductance L.
It depends on:
The square of the number of turns of the coil w
magnetic permeability µ
coil section S
coil length l
L=(w 2 μS)/l , Hn(Henry)
EMF of self-induction:
e L \u003d -Δi / Δt, V
Where Δi/Δt is the rate of current change.
This EMF, preventing a change in current, prevents it from flowing, and therefore creates resistance to alternating current.
Switching surges.
These are overvoltages in circuits with high switching inductance. As a result, an electric arc or spark may occur, the contacts will melt. Therefore, arc extinguishing measures are applied.
Mutual induction.
Mutual induction emf- this is the EMF that occurs in the coil when it is crossed by the changing magnetic flux of another coil.
The transformer works on this principle.
Induced voltage - this is the voltage that occurs in metal structures as a result of their intersection with an alternating magnetic field created by alternating current.
Thus, due to the magnetic field, three types of EMF arise:
1. EMF induction. Occurs when the conductor moves in a constant magnetic field, or when the field moves relative to the conductor.
2. EMF self-induction. Occurs due to the crossing of the conductor by its own changing magnetic field.
3. Mutual induction emf. Occurs when a conductor is crossed by someone else's changing magnetic field.
Eddy currents.
In another way: Foucault currents, induction currents.
These are currents that occur in massive steel parts of electrical installations (cores, cases), due to their intersection with a changing magnetic flux and EMF induction. As a result of the low resistance, the resulting short-circuit currents heat up the machines strongly.
Eddy current losses are power losses that go to heating.
To reduce losses, reduce eddy currents as follows:
1. The cores of electric machines are laminated, that is, they are assembled from sheets of electrical steel insulated with varnish. Thus, the cross section is reduced, which means that the resistance to current is increased.
2. Silicon, which has great resistance, is added to steel.