Induction emf of a moving conductor formula. Magnitude and direction of induction emf
>> induced emf in moving conductors
§ 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:
Lesson content lesson notes supporting frame lesson presentation acceleration methods interactive technologies Practice tasks and exercises self-test workshops, trainings, cases, quests homework discussion questions rhetorical questions from students Illustrations audio, video clips and multimedia photographs, pictures, graphics, tables, diagrams, humor, anecdotes, jokes, comics, parables, sayings, crosswords, quotes Add-ons abstracts articles tricks for the curious cribs textbooks basic and additional dictionary of terms other Improving textbooks and lessonscorrecting errors in the textbook updating a fragment in a textbook, elements of innovation in the lesson, replacing outdated knowledge with new ones Only for teachers perfect lessons calendar plan for the year; methodological recommendations; discussion program Integrated LessonsA straight conductor AB moves in a magnetic field with induction B along conductive buses that are closed to a galvanometer.
Electric charges moving with a conductor in a magnetic field are subject to the Lorentz force:
Fl = /q/vB sin a
Its direction can be determined by the left-hand rule.
Under the influence 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 an external force, and an induced emf occurs in the conductor, and a potential difference arises at the ends of the conductor AB.
The reason for the occurrence of induced 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 induced current appear 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 an induced current occur 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
EMF is an abbreviation of three words: electromotive force. Induction emf () appears in a conducting body that is in an alternating magnetic field. If a conducting body is, for example, a closed circuit, then an electric current flows in it, which is called induction current.
Faraday's law for electromagnetic induction
The main law that is used in calculations related to electromagnetic induction is Faraday's law. He says that the electromotive force of electromagnetic induction in a circuit is equal in magnitude and opposite in sign to the rate of change of magnetic flux () through the surface that is limited by the circuit in question:
Faraday's law (1) is written for the SI system. It must be taken into account that from the end of the normal vector to the contour, the circuit must be traversed counterclockwise. If the flux changes uniformly, then the induced emf is found as:
The magnetic flux that covers the conductive circuit can change due to various reasons. This could be a time-varying magnetic field, deformation of the circuit itself, or movement of the circuit in the field. The total derivative of the magnetic flux with respect to time takes into account the action of all causes.
Induction emf in a moving conductor
Let us assume that a conducting circuit moves in a constant magnetic field. Induction emf occurs in all parts of the circuit that intersect the magnetic field lines. In this case, the resulting EMF appearing in the circuit will be equal to the algebraic sum of the EMF of each section. The occurrence of EMF in the case under consideration is explained by the fact that any free charge that moves along with a conductor in a magnetic field will be acted upon by the Lorentz force. When exposed to Lorentz forces, charges move and form an induction current in a closed conductor.
Consider the case when there is a rectangular conducting frame in a uniform magnetic field (Fig. 1). One side of the frame can move. The length of this side is l. This will be our moving guide. Let's determine how to calculate the induced emf in our conductor if it moves with speed v. The magnitude of the magnetic field induction is B. The plane of the frame is perpendicular to the magnetic induction vector. The condition is met.
The induced emf in the circuit we are considering will be equal to the emf that arises only in its moving part. There is no induction in the stationary parts of the circuit in a constant magnetic field.
To find the induced emf in the frame, we will use the basic law (1). But first, let's define magnetic flux. By definition, the magnetic induction flux is equal to:
where, since by condition the plane of the frame is perpendicular to the direction of the field induction vector, therefore, the normal to the frame and the induction vector are parallel. The area enclosed by the frame can be expressed as follows:
where is the distance over which the moving conductor moves. Let's substitute expression (2), taking into account (3) into Faraday's law, we get:
where v is the speed of movement of the moving side of the frame along the X axis.
If the angle between the direction of the magnetic induction vector () and the velocity vector of the conductor () is an angle , then the EMF module in the conductor can be calculated using the formula:
Examples of problem solving
EXAMPLE 1
| Exercise | Obtain an expression for determining the induced emf modulus in a conductor of length l, which moves in a uniform magnetic field, using the expression for the Lorentz force. The conductor in Fig. 2 moves at a constant speed, parallel to itself. The vector is perpendicular to the conductor and makes an angle with the direction. |
| Solution | Let's consider the force with which the magnetic field acts on a charged particle moving at speed, we get: The work done by the Lorentz force on path l will be: Induction emf can be defined as the work done to move a unit positive charge:
|
| Answer |
EXAMPLE 2
| Exercise | The change in the magnetic flux through the circuit of a conductor having a resistance Ohm for a time equal to s amounted to Wb. What is the current strength in the conductor if the change in magnetic flux can be considered uniform? |
| Solution | With a uniform change in magnetic flux, the basic law of electromagnetic induction can be written as: |
The magnetic flux through the circuit may change for the following reasons:
- When placing a stationary conducting circuit in an alternating magnetic field.
- When a conductor moves in a magnetic field, which may not change over time.
In both of these cases, the law of electromagnetic induction will be fulfilled. Moreover, the origin of the electromotive force in these cases is different. Let's take a closer look at the second of these cases.
In this case, the conductor moves in a magnetic field. Together with the conductor, all the charges that are inside the conductor also move. Each of these charges will be affected by the Lorentz force from the magnetic field. It will promote the movement of charges inside the conductor.
- Induction emf in this case will be of magnetic origin.
Consider the following experiment: a magnetic circuit, one side of which is movable, is placed in a uniform magnetic field. The moving side of length l begins to slide along the sides MD and NC at a constant speed V. At the same time, it constantly remains parallel to the side CD. The magnetic induction vector of the field will be perpendicular to the conductor and make an angle a with the direction of its speed. The following figure shows the laboratory setup for this experiment:
The Lorentz force acting on a moving particle is calculated using the following formula:
Fl = |q|*V*B*sin(a).
The Lorentz force will be directed along the segment MN. Let's calculate the work of the Lorentz force:
A = Fl*l = |q|*V*B*l*sin(a).
Induction emf is the ratio of the work done by a force when moving a unit positive charge to the magnitude of this charge. Therefore, we have:
Ei = A/|q| = V*B*l*sin(a).
This formula will be valid for any conductor moving at a constant speed in a magnetic field. The induced emf will be only in this conductor, since the remaining conductors of the circuit remain stationary. Obviously, the induced emf in the entire circuit will be equal to the induced emf in the moving conductor.
EMF from the law of electromagnetic induction
The magnetic flux through the same circuit as in the example above will be equal to:
Ф = B*S*cos(90-a) = B*S*sin(a).
Here angle (90-a) = angle between the magnetic induction vector and the normal to the contour surface. Over some time ∆t, the contour area will change by ∆S = -l*V*∆t. The minus sign indicates that the area is decreasing. During this time, the magnetic flux will change:
∆Ф = -B*l*V*sin(a).
Then the induced emf is equal to:
Ei = -∆Ф/∆t = B*l*V*sin(a).
If the entire circuit moves inside a uniform magnetic field at a constant speed, then the induced emf will be zero, since there will be no change in the magnetic flux.
