Induction emf in a moving conductor. Induction emf in moving conductors formula for induction emf in a translationally moving conductor

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§ 13 INDUCTION EMF IN MOVING CONDUCTORS

Let us now consider the second case of the occurrence of an induction current.

When a conductor moves, its free charges move with it. Therefore, the Lorentz force acts on the charges from the magnetic field. It is this that causes the movement of charges inside the conductor. The induced emf is therefore of magnetic origin.

In many power plants around the globe, it is the Lorentz force that causes the movement of electrons in moving conductors.

Let's calculate the induced emf that occurs in a conductor moving in a uniform magnetic field (Fig. 2.10). Let the side of the contour MN of length l slide at a constant speed along the sides NC and MD, remaining parallel to the side CD all the time. The magnetic induction vector of a uniform field is perpendicular to the conductor and makes an angle with the direction of its speed.

The force with which the magnetic field acts on a moving charged particle is equal in magnitude

This force is directed along the conductor MN. The work of the Lorentz force 1 on the path l is positive and amounts to:

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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.

After clarifying the nature of the induced emf that occurs in a stationary conductor located in a changing magnetic field, we learned about the properties of the electric field, which differs from that created by point charges. We also learned that work along a closed loop in a field created by point charges is zero, but in a vortex field it is not zero. It is this field that causes EMF in the conductor. However, if the conductor moves in a constant magnetic field, a potential difference will arise at the ends of the conductor, and an EMF will also arise there. But the nature of this force will be different. In this lesson we will find out the nature of EMF in a conductor moving in a magnetic field.

Subject:Electromagnetic induction

Lesson:Movement of a conductor in a magnetic field

In order to establish the nature of the force in a conductor that moves in a magnetic field, we will conduct an experiment. Let us assume that in a vertical uniform magnetic field with induction () there is a horizontal conductor of length ( l), which moves at a constant speed () perpendicular to the magnetic induction vector of the magnetic field. If we connect a sensitive voltmeter to the ends of this conductor, we will see that it will show the presence of a potential difference at the ends of this conductor. Let's find out where this tension comes from. In this case, there is no loop and no changing magnetic field, so we cannot say that the movement of electrons in the conductor arose as a result of the appearance of a vortex electric field. When the conductor moves as a single whole (Fig. 1), the charges of the conductor and the positive ions that are located in the nodes of the crystal lattice, and free electrons, have a speed of directed movement.

Rice. 1

These charges will be acted upon by the Lorentz force from the magnetic field. According to the “left hand” rule: four fingers located in the direction of movement, turn the palm so that the magnetic induction vector enters the back side, then the thumb will indicate the action of the Lorentz force on positive charges.

The Lorentz force acting on charges is equal to the product of the modulus of the charge that it transfers, multiplied by the modulus of magnetic induction, by the speed and the sine of the angle between the magnetic induction vector and the velocity vector.

This force will do work to transfer electrons over short distances along the conductor.

Then the total work done by the Lorentz force along the conductor will be determined by the Lorentz force multiplied by the length of the conductor.

The ratio of the work done by an external force to move a charge to the amount of charge transferred, as determined by EMF.

(4)

So, the nature of the occurrence of induced emf is the work of the Lorentz force. However, formula 10.4. can be obtained formally, based on the definition of the EMF of electromagnetic induction, when a conductor moves in a magnetic field, crossing lines of magnetic induction, covering a certain area, which can be defined as the product of the length of the conductor and the displacement, which can be expressed in terms of the speed and time of movement. The induced emf in magnitude is equal to the ratio of the change in magnetic flux to time.

The magnetic induction module is constant, but the area that covers the conductor changes.

After substitution, the expressions in formula 10.5. and the abbreviations we get:

The Lorentz force acting along the conductor, due to which the redistribution of charges occurs, is only one component of the forces. There is also a second component, which arises precisely as a result of the movement of charges. If electrons begin to move along a conductor, and the conductor is in a magnetic field, then the Lorentz force begins to act, and it will be directed against the movement of the conductor’s speed. Thus, the summing Lorentz force will be equal to zero.

The resulting expression for the induced emf that occurs when a conductor moves in a magnetic field can also be obtained formally, based on the definition. The induced emf is equal to the rate of change of magnetic flux per unit time, taken with a minus sign.

When a stationary conductor is in a changing magnetic field and when the conductor itself moves in a constant magnetic field, the phenomenon occurs electromagneticinduction. In both cases, an induced emf occurs. However, the nature of this force is different.

1. Kasyanov V.A., Physics 11th grade: Textbook. for general education institutions. - 4th ed., stereotype. - M.: Bustard, 2004. - 416 pp.: ill., 8 l. color on
2. Tikhomirova S.A., Yarovsky B.M., Physics 11. - M.: Mnemosyne.
3. Gendenstein L.E., Dick Yu.I., Physics 11. - M.: Mnemosyne.

1. Fizportal.ru ().
2. Eduspb.com ().
3. Cool physics ().

Homework

1. Kasyanov V.A., Physics 11th grade: Textbook. for general education institutions. - 4th ed., stereotype. - M.: Bustard, 2004. - 416 pp.: ill., 8 l. color on, st. 115, z. 1, 3, 4, art. 133, z. 4.
2. A vertical metal rod 50 cm long moves horizontally at a speed of 3 m/s in a uniform magnetic field with an induction of 0.15 Tesla. The magnetic field induction lines are directed horizontally at right angles to the direction of the rod's velocity vector. What is the induced emf in the rod?
3. At what minimum speed must a rod 2 m long be moved in a uniform magnetic field with a magnetic induction of 50 mT in order for an induced emf of 0.6 V to arise in the rod?
4. * A square made of a 2 m long wire moves in a uniform magnetic field with an induction of 0.3 Tesla (Fig. 2). What is the induced emf on each side of the square? Total induced emf in the circuit? υ = 5 m/s, α = 30°.

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.