Purpose of the lesson: to introduce students to the history of the struggle between the concepts of close action and action at a distance; with the shortcomings of theories, introduce the concept of electric field strength, develop the ability to depict electric fields graphically; use the superposition principle to calculate the fields of a system of charged bodies.
During the classes
Checking homework using the independent work method
Option 1
1. Is it possible to create or destroy an electric charge? Why? Explain the essence of the law of conservation of electric charge.
2. There are two bodies in the air that have equal negative electric charges; the bodies repel each other with a force of 0.9 N. The distance between the charges is 8 cm. Calculate the mass of excess electrons in each body, as well as their number.
Solution. m = m0 N = 9.1·10-31·5·1012= 4.5·10-19 (kg); N = √Fr2/k e ; N= 5·1012 (electrons)
Option-2
1 Why do dissimilar bodies become electrified during friction, but homogeneous bodies are not electrified?
Three conductive balls were brought into contact, the first ball had a charge of 1.8 10-8 C, the second had a charge of 0.3 10-8 C, the third ball had no charge. How is the charge distributed between the balls? With what force will two of them interact in a vacuum at a distance of 5 cm from each other?
Solution. q1+q2+q3= 3q; q = (q1+q2+q3)/3q = 0.5·10-8(C)
F= k q2/r2; F= 9·10-5 (H)
Learning new material
1. Discussion of the issue of transferring the effect of one charge to another. Speakers are heard from “supporters” of the theory of short-range action (the field propagates at the speed of light) and the theory of action at a distance (all interactions propagate instantly). Students' performances are accompanied by demonstrations of experiments on the interaction of electrified bodies. Students can ask questions about proponents of one theory or another.
The teacher helps students draw correct conclusions and leads students to form the concept of an electric field.
2. Electric field - a special form of matter that exists independently of us and our knowledge about it.
3. The main property of the electric field- action on electric charges with some force.
| Electrostatic field | The electrostatic field of stationary charges does not change at all and is inextricably linked with the charges that form it. | ||
| Electric field strength: E= F/ q | The ratio of the force with which the electric field acts on a test positive charge to the value of this charge. Vector Ē̄̄̄̄̄ coincides with the direction of the force acting on the positive charge. | ||
| Electric field strength of a point charge. E =q0/4πξ0ξr2 | The electric field strength of a point charge at a certain point in space is directly proportional to the modulus of the charge of the field source and inversely proportional to the square of the distance from the field source to a given point in space. | ||
| Electrostatic field lines | These are lines whose tangents at each point of the field coincide with the direction of the field strength at that point. | ||
| Principle of field superposition: E = E1+E2+E3+… | When fields from several point charges are superimposed, an electrostatic field is formed, the strength of which at any point is equal to the geometric sum of the strengths from each of the component fields. | ||
| Demonstration of experience: “Justification of the principle of superposition of fields” | Hang a “test charge” (foam plate) on a nylon thread. Impact the “test charge” with a charged body. Then bring another charged body and observe its effect on the “test charge”. Remove the first charged body and observe the action of the second charged body. Draw a conclusion. | ||
Independent work with the book.
1. Read the definition of electric field lines in the textbook.
2. Look carefully at Figures 181 – 184, which show examples of tension lines of various charged bodies and systems of bodies.
3. Answer the questions.
A) How is the magnitude of the tension vector displayed in the figures? By what external sign can one distinguish a field with intense action?
B) Where do the electric field lines begin and end?
Q) Are there breaks in the tension lines?
D) How are the electric field lines located relative to the surface of a charged body?
D) In what case can the electric field be considered uniform?
E) Compare the picture of the field lines of a point charge and a uniformly charged ball.
G) Find out using what formula and within what acceptable limits you can calculate the field strength of a conducting ball.
Let's summarize the lesson
Homework: §92 – 94.
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- Purpose of the lesson: to find out the level of theoretical knowledge of students
Subject : Electric field. Electric field strength. Principle of field superposition
The purpose of the lesson: continue the formation of the concept of “electric field”, introduce its main characteristic; study the principle of superposition of electric fields.
During the classes:
1.Organizing moment. Setting the goals and objectives of the lesson.
2.Knowledge test:
Physical dictation
Electrification of bodies. Law of conservation of charge. Coulomb's law
What is the name of the branch of physics that studies stationary charged bodies? /electrostatics/
What interaction exists between charged bodies and particles? /electromagnetic/
What physical quantity determines the electromagnetic interaction? /electric charge/
Does the magnitude of the charge depend on the choice of reference frame? /No/
Can we say that the charge of a system consists of the charges of the bodies included in the system? /Can/
What is the name of the process that leads to the appearance of electrical charges on bodies? /Electrification/
If a body is electrically neutral, does this mean that it contains no electrical charges? /No/
Is it true that in a closed system the algebraic sum of the charges of all bodies in the system remains constant? /Yes/
If the number of charged particles in a closed system has decreased, does this mean that the charge of the entire system has also decreased? /No/
Do we create an electric charge when electrifying? /No/
Can a charge exist independently of a particle? /No/
A body whose total positive charge of particles is equal to the total negative charge of particles is... /Neutral/
How will the force of interaction between charged particles change as the charge of any of these particles increases? /Will increase/
How will the interaction force change when charges move into the medium? /Will decrease/
How will the interaction force change as the distance between charges increases by 3 times? /Will decrease by 9 times/
What is the name of the quantity that characterizes the electrical properties of a medium? /Dielectric constant of the medium/
In what units is electric charge measured? /In pendants/
3.Learning new material
Electric field
The interaction of charges according to Coulomb's law is an experimentally established fact. However, it does not reveal the physical picture of the interaction process itself. And it does not answer the question of how the action of one charge on another occurs.
Faraday gave the following explanation: There is always an electric field around every electric charge. An electric field is a material object that is continuous in space and capable of acting on other electric charges. The interaction of electric charges is the result of the action of the field of charged bodies.
Electric field is a field created by stationary electric charges.
An electric field can be detected if a test (positive) charge is introduced to a given point.
A test point charge is a charge that does not distort the field under study (does not cause a redistribution of charges creating the field).
Electric field properties:
Acts on charges with some force.
The electric field created by a stationary charge, i.e. electrostatic does not change over time.
An electric field is a special type of matter, the movement of which does not obey Newton’s laws of mechanics. This type of matter has its own laws, properties that cannot be confused with anything else in the surrounding world.
Electric field strength
Physical quantity equal to the ratio of the force with which the electric field acts on the test chargeq, to the value of this charge is calledelectric field strength
and is designated 
:
.
The unit of tension is 1N/C or 1V/m.
The electric field and Coulomb force intensity vectors are co-directed.
An electric field whose strength is the same at all points in space is called uniform.
Lines of tension (field lines) – lines whose tangents at each point coincide with the direction of the vector .
In order to use tension lines to characterize not only the direction, but also the intensity value of the electrostatic field, they are drawn with a certain density: the number of tension lines penetrating a unit surface area perpendicular to the tension lines must be equal to the vector modulus .
If the field is created by a point charge, then the intensity lines are radial straight lines emerging from the charge, if it positive, and included in it, if the charge negative.
Principle of field superposition
Experience shows that if an electric charge q electric fields of several sources act simultaneously, then the resulting force turns out to be equal to the sum acting from each field separately.
Electric fields obey the superposition principle:
The strength of the resulting field created by the system of charges is equal to the geometric sum of the field strengths created at a given point by each of the charges separately:
or 
4. Fixing the material
Solving problems from the collection. problems ed. Rymkevich No. 696,697,698
Homework: §92,93,94
Lesson type: problem-developmental
The purpose of the lesson: Create conditions for:
- formation of ideas about the electric field and its effect on the body; electric force and its dependence on the distance between bodies.
- development of communicative competence through the ability to analyze, compare, and draw conclusions;
- fostering tolerance and a conscious attitude towards learning.
Equipment:
- wooden ruler,
- glass and ebonite rod,
- electrostatic sleeves,
- portraits of D. Maxwell, O. Coulon.
Lesson technology: dialogue.
Forms of training: frontal, group, individual, in pairs.
Teaching methods: verbal, practical.
Progress lesson
1. Organizational moment(1 min.)
Experience: The ruler is placed on the back of the chair so that it is in balance. An ebonite charged stick is taken and carried towards the ruler without touching it. The ruler comes out of rest.
2. Updating knowledge.
- How can you explain the results of the experiment?
- Why does the ruler move?
When studying mechanics, we learned that the action of one body on another occurs directly through the interaction of bodies, and in this experiment we do not observe contact, but we observe movement.
- How can we explain the interaction of bodies in this case?
We write down the key words on the board: strength, interaction.
- It can be assumed that a space with special properties exists around a charged body. There is a problem that needs to be solved.
The writing on the board on the left (sign?).
Let's outline the goals of our lesson (students formulate the goal of the lesson, and the teacher specifies it). Experience is shown to resolve the problem. An ebonite rod approaches the calmly hanging cartridge case, and then a glass rod, while the distance between the cartridge case and the charged body changes. The results of the experiment are analyzed by students.
Write on the board:
- Repulsion.
- Attraction.
- What determines the force with which electric bodies interact?
Writing on the board. From distance.
- How do they interact? (students conclude: the closer the distance between the bodies, the stronger the interaction forces and vice versa).
Having looked and analyzed the experiments, we have studied how the interaction of charged bodies occurs, and by what means this interaction occurs, we do not yet know.
Teacher: Many scientists studied the interaction of charged bodies, but M. Faraday and D. Maxwell, O. Coulomb made a special contribution. As a result, it was established that every charged body is surrounded by a special property of matter, which is called an electric field.
So what is this space with special properties through which interaction between charged bodies occurs?
Writing on the board. Electric field.
A supporting outline appears on the board.
Work with a textbook, with reference literature (students give a definition of the electric field, features of the electric field).
3. Systematization of knowledge.
Teacher: today in class we became acquainted with a special type of matter that exists independently of us and our knowledge about it. And this is called an electric field that exists around a charged body and the field of one charge acts on the field of another charge with some force and this force is called electric force (work with reference notes).
Work in groups, in one minute you must find a solution to the problem that will be offered to you.
- K-1. How can you use an electric field near a charged stick to make a piece of cotton float in the air? Show the experience and give an explanation for it.
- K-2. Show the effect of an electric field using available materials and give an explanation.
- K-3. During general cleaning of the house, do we wipe polished surfaces and glass with a dry cloth made of synthetic fabric, and do we wipe those painted with oil paint with a damp cloth? Why do we “feel” differently about cleaning?
And then you need to evaluate your work in class. Knowledge test sheets are provided. Where you should answer the questions. Then you will let your deskmate check your answers, where he will give you a grade.
4. Reflection stage.
Knowledge test sheet
Subject: Electric field. Electric field strength
The purpose of the lesson : 1) Recall the concept of Electric field. Form the concept of electric field strength
Development of logical and abstract thinking, the ability to reason, defend one’s point of view, and draw conclusions.
Nurturing an active life position, developing a scientific worldview.
Equipment : Educational presentation, video, interactive whiteboard
During the classes
1. Introduction . Determining the goals and objectives of the lesson
2. Homework control
Students choose their own answer topic.
Working with the periodic table
How many electrons are included in the water molecule H 2 O (10)
How many electrons are included in the carbon dioxide molecule CO 2 (28)
How many protons are included in the iron oxide molecule Fe 2 O 3 (56)
Charles Coulon's experience
State Coulomb's law
Physical meaning of the proportionality coefficient
Limits of applicability of Coulomb's law
Problems involving the application of Coulomb's law
How will the force of the Coulomb interaction between two point charges change when each charge increases threefold? (increases by 9)
How will the force of interaction between charges change if the distance is reduced by 2 times? (increase by 4 times)
How will the force of the Coulomb interaction of two point charges change when each charge increases threefold, if the distance is reduced by a factor of 2? (increase 36 times)
Two identical metal balls are charged with charges equal in magnitude but opposite in sign. The balls were brought into interaction and moved apart. Determine the force of interaction between charges. (0)
3. Explanation of new material. (Conversation)
We answered the question How charged bodies interact. However, they did not say anything about how the action of one charge on another is carried out.
Let's first discuss the question of how interaction between bodies occurs in general.
1) Theory of action at a distance ( Bodies interact with each other at a distance, and the interaction is transmitted instantly)
2) The theory of short-range action(An intermediate agent is required for interaction to occur)
Which theory is most suitable to describe the interaction of charged bodies?
3) Michael Faraday. ( There is an electric field)
James Maxwell. ( Created the theory of the electromagnetic field)
4) Electric field is a special form of matter
Properties:
Acts on the charge with some force
Generated by electric charges
Detected by its effect on electrical charges
5) Tension – force characteristic of the electric field
Definition: Tension is a physical quantity equal to the ratio of the force with which the electric field acts on a test electric charge to the value of this charge.
Units:(Independently) N/C
Tension vector direction coincides with the direction of the force acting from the electric field on the positive charge
Draw the tension vectors at points A and B
6) Derivation of the formula for the field strength of a point charge. (On one's own)
7) The principle of superposition of fields
8) Electric field strength lines
Lines whose tangents coincide with coincide with the direction of the intensity vector at a given point in the field
9) Properties of electric field lines
§ 91-94
Exercise 17 (1)
Start on positive and end on negative charges
Do not intersect
What new have you learned? (Formulas)
6) Homework
Grading
Unified State Exam discipline section: _________ _
Total lessons in the topic –_18___
№ lesson from this topic _4____
Lesson topic « Electricity. Current strength »
Lesson summary provided
FULL NAME. _ __ Bryleva Liliya Zakirzyanovna_
Academic title, position: Physics teacher
Place of work: Municipal educational institution secondary school No. 6
Physics lesson notes
"Electricity. Current strength."
Lesson objectives:
Educational - give the concept of electric current and find out the conditions under which it occurs. Enter the quantities characterizing the electric current.
Developmental - to form intellectual skills to analyze and compare the results of experiments; activate students’ thinking and ability to draw their own conclusions.
educational - development of cognitive interest in the subject, broadening the horizons of students, showing the possibility of using the knowledge gained in lessons in life situations.
Lesson type: lesson on learning new knowledge.
Equipment: presentation on the topic “Electric current. Current strength."
Lesson plan.
Organizing time.
Updating knowledge.
Learning new material.
Consolidation.
Summarizing.
1. Organizational moment.
Preparation for learning new material.
Today we will get acquainted with the concepts: electric current, current strength and the conditions necessary for the existence of electric current.
3. Updating knowledge.
On the screen is slide number 2.
You all know the phrase “electric current” well, but more often we use the word “electricity”. These concepts have become part of our lives so long ago that we don’t even think about their meaning. So what do they mean?
In previous lessons, we partially touched on this topic, namely, we studied stationary charged bodies. As you remember, this branch of physics is called electrostatics.
On the screen is slide number 3.
Okay, now think about it. What does the word "current" mean?
Movement! This means “electric current”, this is the movement of charged particles. It is this phenomenon that we will study in the following lessons.
In 8th grade, we partially studied this physical phenomenon. Then we said that: “electric current is the directed movement of charged particles.”
Today in the lesson we will consider the simplest case of directional movement of charged particles - direct electric current.
Learning new material.
For the emergence and existence of a constant electric current in a substance, the presence of free charged particles is necessary, the movement of which in a conductor causes the transfer of electric charge from one place to another.
On the screen is slide number 5.
However, if charged particles undergo random thermal motion, such as free electrons in a metal, then charge transfer does not occur, which means there is no electric current.
On the screen is slide number 6.
Electric current occurs only with the ordered (directed) movement of charged particles (electrons or ions).
On the screen – slide number 7.
How to make charged particles move in an orderly manner?
We need a force acting on them in a certain direction. As soon as this force ceases to act, the ordered movement of particles will cease due to the electrical resistance exerted to their movement by ions of the crystal lattice of metals or neutral molecules of electrolytes.
On the screen – slide number 8.
So where does this power come from? We said that charged particles are acted upon by the Coulomb force F = q E (the Coulomb force is equal to the product of the charge and the intensity vector), which is directly related to the electric field.
On the screen is slide number 9.
Typically, it is the electric field inside the conductor that causes and maintains the ordered movement of charged particles. If there is an electric field inside a conductor, then there is a potential difference between the ends of the conductor. When the potential difference does not change over time, a constant electric current is established in the conductor.
On the screen – slide number 10
This means that in addition to charged particles, for the existence of an electric current, the presence of electric field.
When a potential difference (voltage) is created between any points of the conductor, the balance of charges will be disrupted and a movement of charges will occur in the conductor, which is called an electric current.
On the screen – slide number 11.
Thus, we have established two conditions for the existence of electric current:
presence of free charges,
presence of an electric field.
On the screen is slide number 12.
So: ELECTRIC CURRENT is the directed, ordered movement of charged particles (electrons, ions and other charged particles.). Those. electric current has a certain direction. The direction of current is taken to be the direction of movement of positively charged particles. It follows that the direction of the current coincides with the direction of the electric field strength vector. If the current is formed by the movement of negatively charged particles, then the direction of the current is considered opposite to the direction of movement of the particles. (This choice of current direction is not very successful, since in most cases the current represents the ordered movement of electrons - negatively charged particles. The choice of current direction was made at a time when nothing was known about free electrons in metals.)
On the screen is slide number 13.
We do not directly see the movement of particles in a conductor. The presence of electric current must be judged by the actions or phenomena that accompany it.
On the screen is slide number 14.
Thermal effect of electric current. The conductor through which the current flows heats up (an incandescent light bulb lights up);
On the screen is slide number 15.
Magnetic effect of electric current. A conductor with current attracts or magnetizes bodies, turns perpendicular to the wire with current, a magnetic arrow;
On the screen is slide number 16.
Chemical action of electric current. An electric current can change the chemical composition of a conductor, for example, releasing its chemical constituents (hydrogen and oxygen are released from acidified water poured into a U-shaped glass vessel).
The magnetic effect is the main one, as it is observed in all conductors, the thermal effect is absent in superconductors, and the chemical effect is observed only in solutions and melts of electrolytes.
On the screen is slide number 17.
Like many physical phenomena, electric current has a quantitative characteristic called current strength: if through the cross section the conductor carries a charge ∆q during the time ∆t, then the average value of the current is: I=∆q/∆t(current strength is equal to the ratio of charge to time).
Thus, the average current strength is equal to the ratio of the charge ∆q passing through the cross section of the conductor during the time interval ∆t to this period of time.
In the SI (International System) the unit of current is the ampere, denoted 1 A = 1 C/s (One ampere is equal to the ratio of 1 coulomb per 1 second)
Please note: if the current does not change over time, then the current is called constant.
On the screen is slide number 18.
The current strength can be a positive value if the direction of the current coincides with the conventionally selected positive direction along the conductor. Otherwise the current is negative.
On the screen is slide number 19.
To measure current, a device called an ammeter is used. The design principle of these devices is based on the magnetic action of current. An ammeter is connected in an electrical circuit in series to the device from which the current is to be measured. A schematic representation of an ammeter is a circle with the letter A in the center.
On the screen is slide number 20.
In addition, the current strength is related to the speed of directional movement of particles. Let's show this connection.
Let a cylindrical conductor have a cross section S. Let us take the direction from left to right as the positive direction in the conductor. The charge of each particle will be considered equal to q 0. The volume of the conductor, limited by cross sections 1 and 2 with a distance ∆L between them, contains particles N = n·S·∆L, where n is the concentration of particles.
On the screen is slide number 21.
Their total charge in the selected volume is q = q 0 ·n·S·∆L (the charge is equal to the product of the particle charge by concentration, area and distance). If particles move from left to right with an average speed v, then in a time ∆t = ∆L/v equal to the ratio of distance to speed, all particles contained in the volume under consideration will pass through cross section 2. Therefore, the current strength is found using the following formula.
I = ∆q/∆t = (q 0 ·n·S·∆L·v)/∆L= q 0 ·n·S·v
On the screen is slide number 22.
Using this formula, let's try to determine the speed of ordered movement of electrons in a conductor.
V = I/( e·n·S),
Where e– electron charge modulus.
On the screen is slide number 23.
Let the current strength I = 1A, and the cross-sectional area of the conductor S = 10 -6 m 2, for copper the concentration n = 8.5 10 28 m -3. Hence,
V=1/(1.6 ·10 -19 · 8.5·10 28 ·10 -6)=7·10 -5 m/s
As we see, the speed of ordered movement of electrons in a conductor is low.
On the screen is slide number 24.
To estimate how small, n Let us imagine a very long current circuit, for example a telegraph line between two cities separated from each other, say, 1000 km. Careful experiments show that the effects of the current in the second city will begin to manifest themselves, that is, the electrons in the conductors located there will begin to move, approximately 1/300 of a second after their movement along the wires in the first city began. It is often said, not very strictly, but very clearly, that current travels through wires at a speed of 300,000 km/s. This, however, does not mean that the movement of charge carriers in the conductor occurs at this enormous speed, so that an electron or ion, which in our example was in the first city, will reach the second in 1/800 of a second. Not at all. The movement of carriers in a conductor almost always occurs very slowly, at a speed of several millimeters per second, and often even less. We see, therefore, that we need to carefully distinguish and not confuse the concepts of “current speed” and “speed of charge carriers.”
On the screen is slide number 25.
Thus, the speed that we call “current speed” for brevity is the speed of propagation of changes in the electric field along the conductor, and not at all the speed of movement of charge carriers in it.
Let us explain this with a mechanical analogy. Let's imagine that two cities are connected by an oil pipeline and that in one of these cities a pump has begun to operate, increasing the oil pressure in that place. This increased pressure will spread through the liquid in the pipe at high speed - about a kilometer per second. Thus, in a second, particles will begin to move at a distance of, say, 1 km from the pump, after two seconds - at a distance of 2 km, in a minute - at a distance of 60 km, etc. After about a quarter of an hour, oil will begin to flow out of the pipe in the second city. But the movement of the oil particles themselves occurs much more slowly, and several days may pass before any specific oil particles reach from the first city to the second. Returning to electric current, we must say that the “speed of current” (the speed of propagation of the electric field) is similar to the speed of pressure propagation through the oil pipeline, and the “velocity of carriers” is similar to the speed of movement of the particles of the oil itself.
5. Consolidation.
On the screen – slide number 26
Today in class we looked at the basic concept of electrodynamics:
Electricity;
Conditions necessary for the existence of electric current;
Quantitative characteristics of electric current.
On the screen – slide No. 27
Now let's look at solving typical problems:
1. The tile is included in the lighting network. How much electricity flows through it in 10 minutes if the current in the supply cord is 5A?
Solution: Time in SI system 10 minutes = 600s,
By definition, current is equal to the ratio of charge to time.
Hence, the charge is equal to the product of current and time.
Q = I t = 5A 600 s = 3000 C
On the screen – slide No. 28
2. How many electrons pass through the filament of an incandescent lamp in 1 s when the current in the lamp is 1.6 A?
Solution: The charge of an electron is e= 1.6 10 -19 C,
The entire charge can be calculated using the formula:
Q = I t – charge is equal to the product of current and time.
The number of electrons is equal to the ratio of the total charge to the charge of one electron:
N = q/ e
this implies
N = I t / e= 1.6A 1s/1.6 10 -19 Cl = 10 19
On the screen – slide No. 29
3. A current of 1 A flows through a conductor for a year. Find the mass of electrons that passed through the cross section of the conductor during this period of time. Ratio of electron charge to its mass e/m e = 1.76 10 +11 C/kg.
Solution: The mass of electrons can be defined as the product of the number of electrons and the mass of the electron M = N m e. Using the formula N = I t / e(see previous problem), we find that the mass is equal to
М = m e I t / e= 1A 365 24 60 60s/(1.76 10 +11 C/kg) = 1.8 10 -4 kg.
On the screen – slide number 30
4. In a conductor with a cross-sectional area of 1 mm 2, the current is 1.6 A. The electron concentration in the conductor is 10 23 m -3 at a temperature of 20 0 C. Find the average speed of directional movement of electrons and compare it with the thermal speed of electrons.
Solution: To determine the average speed of directional movement of electrons, we use the formula
Q = q 0 n S v t (the charge is equal to the product of the particle charge by concentration, area, speed and time).
Since I = q/t (current strength is equal to the ratio of charge to time),
Then I = q 0 n S v => v= I/ (q 0 n S)
Let us calculate and obtain the value of the speed of electron movement
V= 1.6A/(10 23 m -3 10 -6 m 1.6 10 -19 C) = 100 m/s
M v 2 /2 = (3/ 2) k T => (it follows from here)
= 11500 m/s
The speed of thermal movement is 115 times greater.
Summarizing.
On the screen – slide No. 31
Write down your homework.
V.A.Kasyanov Physics textbook 11th grade. §1,2, problems §2 (1-5).
On the screen – slide number 32.
Thank you for your attention. We wish you success in your independent exercises on this topic!
Abstract checked
Methodologist of the Education Department:_____________________________________________
Expert Council of the Yerevan State Pedagogical University:__________________________________________
Date of:_____________________________________________________________
Signatures:__________________________________________________________
