Occurrence of induced emf in a conductor
If placed in conductor and move it so that during its movement it intersects the field lines, then a, called induced emf.
An induced emf will occur in a conductor even if the conductor itself remains stationary, and the magnetic field moves, crossing the conductor with its lines of force.
If the conductor in which the induced emf is induced is closed to any external circuit, then under the influence of this emf a current called induction current.
The phenomenon of EMF induction in a conductor when it is crossed by magnetic field lines is called electromagnetic induction.
Electromagnetic induction is a reverse process, i.e., the conversion of mechanical energy into electrical energy.
The phenomenon of electromagnetic induction has found wide application in. The design of various electrical machines is based on its use.
Magnitude and direction of induced emf
Let us now consider what the magnitude and direction of the EMF induced in the conductor will be.
The magnitude of the induced emf depends on the number of field lines crossing the conductor per unit time, i.e., on the speed of movement of the conductor in the field.
The magnitude of the induced emf is directly dependent on the speed of movement of the conductor in the 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 larger 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 that appears in the conductor crossing this field.
So, the magnitude of the induced emf that occurs in a 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 induced emf; B - magnetic induction; I is the length of the conductor; v is the speed of movement of the conductor.
It should be firmly remembered that In a conductor moving in a magnetic field, an induced emf occurs only if this conductor is crossed by magnetic field lines. If the conductor moves along the field lines, that is, does not cross, but seems to slide 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 will indicate the direction of action of the induced emf and the direction of the current in the conductor.
Right hand rule
Induction emf in a coil
We have already said that in order to create an inductive emf 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 therefore the induced current, can be obtained not only in a straight conductor, but also in a conductor twisted 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., exactly as it was when a straight conductor moved in the field of the magnet.
If the magnet is lowered into the coil slowly, then the EMF arising in it will be so small that the needle of the device may not even deviate. If, on the contrary, the magnet is quickly inserted into the coil, then the deflection of the needle will be large. This means that the magnitude of the induced emf, and therefore the current strength in the coil, depends on the speed of movement of the magnet, i.e., on how quickly the field lines intersect the turns of the coil. If you now alternately introduce a strong magnet and then a weak one into the coil at the same speed, you will notice that with a strong magnet the needle of the device will deviate at a larger angle. Means, the magnitude of the induced emf, and therefore the current strength in the coil, depends on the magnitude of the magnetic flux of the magnet.
And finally, if you introduce the same magnet at the same speed, first into a coil with a large number of turns, and then with a significantly smaller number, then in the first case the needle of the device will deflect at a larger angle than in the second. This means that the magnitude of the induced emf, and therefore 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 induced emf in the coil depends on the direction of movement of the magnet. The law established by E. H. Lenz tells how to determine the direction of the induced emf.
Lenz's law for electromagnetic induction
Any change in the magnetic flux inside the coil is accompanied by the appearance of an induced emf in it, and the faster the magnetic flux passing through the coil changes, the greater the emf is induced in it.
If the coil in which the induced emf is created is closed to an external circuit, then an induced current flows through its turns, creating a magnetic field around the conductor, due to which the coil turns into a solenoid. It turns out that a changing external magnetic field causes an induced current in the coil, which, in turn, creates its own magnetic field around the coil - the current field.
Studying this phenomenon, E. H. Lenz established a law that determines the direction of the induced current in the coil, and therefore the direction of the induced emf. The induced emf that occurs in a 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 the way in which a change in the external magnetic field is achieved.
When a permanent magnet moves relative to a wire coil connected to the terminals of a galvanometer, or when a coil moves relative to a magnet, an induced current occurs.
Induction currents in massive conductors
A changing magnetic flux is capable of inducing an emf not only in the turns of the coil, but also in massive metal conductors. Penetrating the thickness of a massive conductor, the magnetic flux induces an emf in it, creating induced currents. These so-called ones spread along a massive conductor and short-circuit in it.
The cores of transformers, magnetic circuits of various electrical machines and devices are precisely those massive conductors that are heated by the induction currents arising in them. This phenomenon is undesirable, therefore, to reduce the magnitude of induced currents, parts of electrical machines and transformer cores are not made massive, but consist of thin sheets, isolated from one another with paper or a layer of insulating varnish. Due to this, the path of propagation of eddy currents through the mass of the conductor is blocked.
But sometimes in practice eddy currents are also used as useful currents. For example, the work of so-called magnetic dampers of moving parts of electrical measuring instruments is based on the use of these currents.
The appearance of electromotive force (EMF) in bodies moving in a magnetic field is easy to explain if we recall the existence of the Lorentz force. Let the rod move in a uniform magnetic field with induction Fig. 1. Let the direction of the speed of movement of the rod () and be perpendicular to each other.
Between points 1 and 2 of the rod, an EMF is induced, which is directed from point 1 to point 2. The movement of the rod is the movement of positive and negative charges that are part of the molecules of this body. The charges move together with the body in the direction of movement of the rod. The magnetic field affects the charges using the Lorentz force, trying to move positive charges towards point 2, and negative charges towards the opposite end of the rod. Thus, the action of the Lorentz force generates an induced emf.
If a metal rod moves in a magnetic field, then positive ions, located at the nodes of the crystal lattice, cannot move along the rod. In this case, mobile electrons accumulate in excess at the end of the rod near point 1. The opposite end of the rod will experience a shortage of electrons. The voltage that appears determines the induced emf.
If the moving rod is made of a dielectric, the separation of charges under the influence of the Lorentz force leads to its polarization.
The induced emf will be zero if the conductor moves parallel to the direction of the vector (that is, the angle between and is zero).
Induction emf in a straight conductor moving in a magnetic field
Let us obtain a formula for calculating the induced emf that occurs in a straight conductor of length l moving parallel to itself in a magnetic field (Fig. 2). Let v be the instantaneous speed of the conductor, then in time it will describe an area equal to:
In this case, the conductor will cross all the lines of magnetic induction that pass through the pad. We obtain that the change in magnetic flux () through the circuit into which the moving conductor enters:
where is the component of magnetic induction perpendicular to the area. Let us substitute the expression for (2) into the basic law of electromagnetic induction:
In this case, the direction of the induction current is determined by Lenz's law. That is, the induction current has such a direction that the mechanical force that acts on the conductor slows down the movement of the conductor.
Induction emf in a flat coil rotating in a magnetic field
If a flat coil rotates in a uniform magnetic field, the angular velocity of its rotation is equal to , the axis of rotation is in the plane of the coil and , then the induced emf can be found as:
where S is the area limited by the coil; - coil self-induction flux; - angular velocity; () - angle of rotation of the contour. It should be noted that expression (5) is valid when the axis of rotation makes a right angle with the direction of the external field vector.
If the rotating frame has N turns and its self-induction can be neglected, then:
Examples of problem solving
EXAMPLE 1
| Exercise | A car antenna located vertically moves from east to west in the Earth's magnetic field. The antenna length is m, the moving speed is . What will be the voltage between the ends of the conductor? |
| Solution | The antenna is an open conductor, therefore, there will be no current in it, the voltage at the ends is equal to the induced emf: The component of the magnetic induction vector of the Earth's field perpendicular to the direction of motion of the antenna for mid-latitudes is approximately equal to T. |
When a straight conductor moves in a magnetic field, e.m. 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. Let us equate formulas (1) and (2) § 97:
BIls = EIΔt, from here
Where s/Δt=v is the speed of movement of the conductor. Therefore e. d.s. induction when the conductor moves perpendicular to the magnetic field lines
E = Blv.
If the conductor moves with 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 induction lines and does not cause emission in it when the conductor moves. d.s. induction. In the conductor e. d.s. is induced only due to the component v 2 = v sin α, directed perpendicular to the induction lines. In this case e. d.s. induction will be
E = 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, an 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, an e will appear. d.s. on its two sides (see Fig. 138). In this case e. d.s. induction will be E = 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 occurs in it. d.s. induction
E = 2nBlv sin α.
What uh. d.s. induction depends on the speed v of rotation of the frame and on the induction B of the magnetic field, which can be seen in this experiment (Fig. 148, b). When the armature of the current generator rotates slowly, the light bulb lights up dimly: a low emission has occurred. d.s. induction. As the speed of rotation of the armature increases, the light bulb burns brighter: a larger e occurs. d.s. induction. At the same speed of armature rotation, we remove one of the magnets, thereby reducing the magnetic field induction. The light is dimly lit: eh. d.s. induction decreased.
Problem 35. Straight conductor length 0.6 m connected to a current source by flexible conductors, e.g. d.s. whom 24 V and internal resistance 0.5 ohm. The conductor is in a uniform magnetic field with induction 0.8 tl, the induction lines of which are directed towards the reader (Fig. 149). Resistance of the entire external circuit 2.5 ohm. Determine the current strength in the conductor if it moves perpendicular to the induction lines at speed 10 m/sec. What is the current strength in a stationary conductor?
MOVING IN THE FIELD
In modern machines - generators - the generation of EMF is based on the law just discussed. However, unlike the examples in the previous paragraph, in electric machines, a change in magnetic flux occurs due to the movement of a conductor in a magnetic field.
Let's imagine that in a narrow gap between the poles of a large electromagnet there is part of a rigid rectangular frame bent from a thick wire (Fig. 2.28 and 2.29). This frame is not completely closed, and its ends are connected with a flexible Cord. The cord is connected to the galvanometer. When the frame moves in the direction indicated by the arrow, the magnetic flux coupled to the frame will change. When the magnetic flux changes, an emf is induced. The magnitude of the EMF can be judged by the deflection of the galvanometer needle.
Rice. 2.28. A frame made of rigid wire is pushed into the gap between the poles of the electromagnet. The frame circuit is closed by wires connected to the galvanometer
Rice. 2.29. Same as in fig. 2.28, but for clarity the top of the electromagnet (south pole) is not shown. Arrow v shows the direction of movement of the frame. The width of the frame is indicated by the letter I. Dimension a shows how deep the frame is pushed into the slot. The magnetic field is shown by a series of arrows
In Fig. 2.29, for clarity of the figure, the upper part of the electromagnet (south pole) is not shown at all. In the same figure, the magnetic field is depicted by a series of small arrows. The field between the poles is directed exactly as shown by the small arrows. In the space between the poles the field has a constant induction. As you move away from the poles, the field weakens very quickly. One can even safely assume that there is no field outside the gap.
Let's calculate the magnetic flux Ф covered by the frame.
To do this, you need to multiply the magnetic induction B by that part of the frame area that is located between the poles.
If the frame has a width I and is extended to a depth a (Fig. 2.29), then the area S penetrated by the field is
Magnetic flux coupled to the frame
The deeper the frame is retracted, the greater the flow.
Let the frame reach the middle of the pole width as shown in the picture. In this case, the flow linked to it is depicted by 16 lines. Let's move the frame even deeper, so that it reaches 3/4 of the width of the pole. Then the stream will already consist of 24 lines. When the frame covers the entire pole, the flow will increase to 32 lines.
But what is the rate of increase in flow?
It, of course, depends on the speed with which the frame moves into the gap between the poles.
But it is possible to more accurately determine the rate of increase in flow.
When moving the frame in the formula
only the size a changes (the depth to which the frame is retracted), which means that the change in the AF flux depends on the change in this particular size a.
Over a period of time, the increase in this size can be represented by the following formula:
where is the speed at which the frame moves.
But if we know the change in size a (i.e.), then it is not difficult to calculate the corresponding change in flow ():
Thus, we are almost finished deriving the formula for the induced emf. We only need to determine the rate of change of the flow. Dividing the left and right sides of the last equality by we find
This is the formula for calculating the EMF,
induced in a straight conductor moving in a magnetic field at a speed
The derived formula is valid when: 1) the conductor is located at right angles to the direction of the magnetic field and to the direction of speed and 2) the speed also forms a right angle to the direction of the field.
The conclusions presented here are also valid in the case when the wire is stationary, and the poles themselves move along with the magnetic field they create.
We found a formula for the motion of the frame, and applied it as a formula for the emf induced in a straight conductor moving across the field. It is easy to explain the reasons for this: in the side wires located parallel to the direction of speed, no EMF is induced. The entire emf is induced in a transverse wire of length l moving in a magnetic field.
In fact, if this transverse wire goes beyond the field, then with further movement of the frame, the flow coupled with it will reach its maximum value (32 lines) and will not change. Of course, only until the back side of the frame fits into the gap between the poles. This means that no EMF is induced in the side wires (parallel), even when they move in a magnetic field.
Rice. 2.30. Right hand rule
Right hand rule. The direction of the EMF induced when the wire moves can be determined using the right-hand rule (Fig. 2.30):
if the right hand is positioned so that the field lines enter the palm, and the bent thumb coincides with the direction of movement, then the four extended fingers show the direction of the induced emf.
The direction of the induced EMF is the direction in which current should flow in a closed circuit under its action.
It is easy to verify that the right-hand rule is completely consistent with Lenz's rule. We leave it to the reader to see for themselves.
Example. A wire moves between the poles, as shown in Fig. 2.28 and 2.29. Magnetic induction 1.2 Tesla. Wire length. Speed Find the emf induced in the wire.
Solution. According to the formula
Of course, such an EMF is induced in the wire only during the period of time when the wire is between the poles.
Magnetic fields, speeds and dimensions similar to those shown in this example can be found in electrical machines.
The relationship between electrical and magnetic phenomena has always interested physicists. English physicist Michael Faraday was completely confident in the unity of electrical and magnetic phenomena. He reasoned that an electric current could magnetize a piece of iron. Couldn't a magnet in turn cause an electric current? This problem has been solved.
If a conductor moves in a constant magnetic field, then the free electric charges inside it also move (they are acted upon by the Lorentz force). Positive charges are concentrated at one end of the conductor (wire), negative charges at the other. A potential difference arises - EMF electromagnetic induction. The phenomenon of the occurrence of induced emf in a conductor moving in a constant magnetic field is called phenomenon of electromagnetic induction.
Rule for determining the direction of induction current (right hand rule):
An induced emf occurs in a conductor moving in a magnetic field; the current energy in this case is determined by the Joule-Lenz law:
Work done by an external force to move a current-carrying conductor in a magnetic field
Induction EMF in the circuit
Let's consider the change in magnetic flux through a conducting circuit (coil). The phenomenon of electromagnetic induction was discovered experimentally:
Law of electromagnetic induction (Faraday's law): The electromagnetic induction emf arising in the circuit is directly proportional to the rate of change of the magnetic flux through it.
