>> Analogy between mechanical and electromagnetic oscillations

§ 29 ANALOGY BETWEEN MECHANICAL AND ELECTROMAGNETIC OSCILLATIONS

Electromagnetic oscillations in the circuit are similar to free mechanical oscillations, for example, to oscillations of a body fixed on a spring (spring pendulum). The similarity does not refer to the nature of the quantities themselves, which change periodically, but to the processes of periodic change of various quantities.

During mechanical vibrations, the coordinate of the body periodically changes X and the projection of its speed x, and with electromagnetic oscillations, the charge q of the capacitor and the current strength change i in the chain. The same nature of the change in quantities (mechanical and electrical) is explained by the fact that there is an analogy in the conditions under which mechanical and electromagnetic oscillations occur.

The return to the equilibrium position of the body on the spring is caused by the elastic force F x control, proportional to the displacement of the body from the equilibrium position. The proportionality factor is the spring constant k.

The discharge of the capacitor (appearance of current) is due to the voltage between the plates of the capacitor, which is proportional to the charge q. The coefficient of proportionality is the reciprocal of the capacitance, since u = q.

Just as, due to inertia, a body only gradually increases its speed under the action of forces, and this speed does not immediately become equal to zero after the force ceases to act, electricity in the coil, due to the phenomenon of self-induction, increases gradually under the influence of voltage and does not disappear immediately when this voltage becomes equal to zero. The circuit inductance L plays the same role as the body mass m during mechanical vibrations. Accordingly, the kinetic energy of the body is similar to the energy magnetic field current

Charging a capacitor from a battery is similar to communicating potential energy to a body attached to a spring when the body is displaced by a distance x m from the equilibrium position (Fig. 4.5, a). Comparing this expression with the energy of the capacitor, we notice that the stiffness k of the spring plays the same role during mechanical vibrations as the reciprocal of the capacitance during electromagnetic vibrations. In this case, the initial coordinate x m corresponds to the charge q m .

The appearance of current i in the electric circuit corresponds to the appearance of body speed x in the mechanical oscillatory system under the action of the elastic force of the spring (Fig. 4.5, b).

The moment in time when the capacitor is discharged and the current strength reaches its maximum is similar to the moment in time when the body passes at maximum speed (Fig. 4.5, c) the equilibrium position.

Further, the capacitor in the course of electromagnetic oscillations will begin to recharge, and the body, in the course of mechanical oscillations, will begin to shift to the left from the equilibrium position (Fig. 4.5, d). After half the period T, the capacitor will be fully recharged and the current will become zero.

With mechanical vibrations, this corresponds to the deviation of the body to the extreme left position, when its speed is zero (Fig. 4.5, e).

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§ 29. Analogy between mechanical and electromagnetic oscillations

During mechanical vibrations, the coordinate of the body periodically changes X and the projection of its speed v x, and with electromagnetic oscillations, the charge changes q capacitor and current i in the chain. The same nature of the change in quantities (mechanical and electrical) is explained by the fact that there is an analogy in the conditions under which mechanical and electromagnetic oscillations occur.

Just as, due to inertia, the body only gradually increases its speed under the action of a force, and this speed does not immediately become equal to zero after the termination of the force, the electric current in the coil, due to the phenomenon of self-induction, increases gradually under the action of voltage and does not disappear immediately when this voltage becomes equal to zero. The loop inductance L plays the same role as the mass of the body m during mechanical vibrations. Accordingly, the kinetic energy of the body is similar to the energy of the magnetic field of the current

Charging a capacitor from a battery is similar to communicating a body attached to a spring with potential energy when the body is displaced by a distance x m from the equilibrium position (Fig. 4.5, a). Comparing this expression with the energy of the capacitor, we notice that the stiffness k of the spring plays the same role during mechanical vibrations as the reciprocal of the capacitance during electromagnetic vibrations. In this case, the initial coordinate x m corresponds to the charge q m .

The appearance of current i in the electric circuit corresponds to the appearance of the body speed v x in the mechanical oscillatory system under the action of the elastic force of the spring (Fig. 4.5, b).

With mechanical vibrations, this corresponds to the deviation of the body to the extreme left position, when its speed is zero (Fig. 4.5, e). The correspondence between mechanical and electrical quantities during oscillatory processes can be summarized in a table.

Electromagnetic and mechanical vibrations are of different nature, but are described by the same equations.

Questions for the paragraph

1. What is the analogy between electromagnetic oscillations in a circuit and oscillations of a spring pendulum?

2. Due to what phenomenon does the electric current in the oscillatory circuit not immediately disappear when the voltage across the capacitor becomes zero?

Development of a methodology for studying the topic "Electromagnetic oscillations"

Oscillatory circuit. Energy transformations during electromagnetic oscillations.

These questions, which are among the most important in this topic, are dealt with in the third lesson.

First, the concept of an oscillatory circuit is introduced, an appropriate entry is made in a notebook.

Further, in order to find out the cause of the occurrence of electromagnetic oscillations, a fragment is shown, which shows the process of charging the capacitor. The attention of students is drawn to the signs of the charges of the capacitor plates.

After that, the energies of the magnetic and electric fields are considered, the students are told about how these energies and the total energy in the circuit change, the mechanism for the occurrence of electromagnetic oscillations is explained using the model, and the basic equations are recorded.

It is very important to draw students' attention to the fact that such a representation of the current in the circuit (the flow of charged particles) is conditional, since the speed of electrons in the conductor is very low. This method of representation was chosen to facilitate understanding of the essence of electromagnetic oscillations.

Further, the attention of students is focused on the fact that they observe the processes of energy transformation electric field into magnetic energy and vice versa, and since the oscillatory circuit is ideal (there is no resistance), the total energy of the electromagnetic field remains unchanged. After that, the concept of electromagnetic oscillations is given and it is stipulated that these oscillations are free. Then the results are summed up and homework is given.

Analogy between mechanical and electromagnetic oscillations.

This question is considered in the fourth lesson of the study of the topic. First, for repetition and consolidation, you can once again demonstrate the dynamic model of an ideal oscillatory circuit. To explain the essence and prove the analogy between electromagnetic oscillations and oscillations of a spring pendulum, the dynamic oscillatory model “Analogy between mechanical and electromagnetic oscillations” and PowerPoint presentations are used.

A spring pendulum (oscillations of a load on a spring) is considered as a mechanical oscillatory system. Identification of the relationship between mechanical and electrical quantities in oscillatory processes is carried out according to the traditional method.

As it was already done in the last lesson, it is necessary to remind the students once again about the conditionality of the movement of electrons along the conductor, after which their attention is drawn to the upper right corner of the screen, where the “communicating vessels” oscillatory system is located. It is stipulated that each particle oscillates around the equilibrium position, therefore, fluid oscillations in communicating vessels can also serve as an analogy for electromagnetic oscillations.

If there is time left at the end of the lesson, then you can dwell on the demonstration model in more detail, analyze all the main points using the newly studied material.

Equation of free harmonic vibrations in contour.

At the beginning of the lesson, dynamic models of an oscillatory circuit and analogies of mechanical and electromagnetic oscillations are demonstrated, the concepts of electromagnetic oscillations, an oscillatory circuit, the correspondence of mechanical and electromagnetic quantities in oscillatory processes are repeated.

The new material must begin with the fact that if the oscillatory circuit is ideal, then its total energy remains constant over time

those. its time derivative is constant, and hence the time derivatives of the energies of the magnetic and electric fields are also constant. Then, after a series of mathematical transformations, they come to the conclusion that the equation of electromagnetic oscillations is similar to the equation of oscillations of a spring pendulum.

Referring to the dynamic model, students are reminded that the charge in the capacitor changes periodically, after which the task is to find out how the charge, the current in the circuit and the voltage across the capacitor depend on time.

These dependencies are found by the traditional method. After the equation for the oscillations of the capacitor charge is found, the students are shown a picture that shows the graphs of the charge of the capacitor and the displacement of the load versus time, which are cosine waves.

In the course of elucidating the equation for oscillations of the charge of a capacitor, the concepts of the period of oscillations, cyclic and natural frequencies of oscillations are introduced. Then the Thomson formula is derived.

Next, the equations for fluctuations in the current strength in the circuit and the voltage on the capacitor are obtained, after which a picture is shown with graphs of the dependence of three electrical quantities on time. Students' attention is drawn to the phase shift between current fluctuations and charges by its absence between voltage and charge fluctuations.

After all three equations are derived, the concept of damped oscillations is introduced and a picture is shown showing these oscillations.

On next lesson are summed up summary with the repetition of the basic concepts, the problems of finding the period, cyclic and natural frequencies of oscillations are solved, the dependences q(t), U(t), I(t), as well as various qualitative and graphic problems are investigated.

4. Methodical development three lessons

The lessons below are designed as lectures, since this form, in my opinion, is the most productive and leaves enough time in this case to work with dynamic demos. ion models. If desired, this form can be easily transformed into any other form of the lesson.

Lesson topic: Oscillatory circuit. Energy transformations in an oscillatory circuit.

Explanation of new material.

The purpose of the lesson: explanation of the concept of an oscillatory circuit and the essence of electromagnetic oscillations using the dynamic model “Ideal oscillatory circuit”.

Oscillations can occur in a system called an oscillatory circuit, consisting of a capacitor with a capacitance C and an inductance coil L. An oscillatory circuit is called ideal if there is no energy loss in it for heating the connecting wires and coil wires, i.e., the resistance R is neglected.

Let's make a drawing of a schematic image of an oscillatory circuit in notebooks.

In order for electrical oscillations to occur in this circuit, it is necessary to inform it of a certain amount of energy, i.e. charge the capacitor. When the capacitor is charged, the electric field will be concentrated between its plates.

(Let's follow the process of charging the capacitor and stop the process when the charging is completed).

So, the capacitor is charged, its energy is equal to

therefore, therefore,

Since after charging the capacitor will have a maximum charge (pay attention to the capacitor plates, they have charges opposite in sign), then at q \u003d q max, the energy of the electric field of the capacitor will be maximum and equal to

IN initial moment time, all the energy is concentrated between the plates of the capacitor, the current in the circuit is zero. (Let's now close the capacitor to the coil on our model). When the capacitor closes to the coil, it begins to discharge and a current will appear in the circuit, which, in turn, will create a magnetic field in the coil. The lines of force of this magnetic field are directed according to the gimlet rule.

When the capacitor is discharged, the current does not immediately reach its maximum value, but gradually. This is because the alternating magnetic field generates a second electric field in the coil. Due to the phenomenon of self-induction, an induction current arises there, which, according to the Lenz rule, is directed in the direction opposite to the increase in the discharge current.

When the discharge current reaches its maximum value, the energy of the magnetic field is maximum and is equal to:

and the energy of the capacitor at this moment is zero. Thus, through t=T/4 the energy of the electric field has completely passed into the energy of the magnetic field.

(Let's observe the process of discharging a capacitor on a dynamic model. I draw your attention to the fact that this way of representing the processes of charging and discharging a capacitor in the form of a flow of running particles is conditional and is chosen for ease of perception. You know perfectly well that the speed of electrons is very small ( of the order of several centimeters per second). So, you see how, with a decrease in the charge on the capacitor, the current strength in the circuit changes, how the energies of the magnetic and electric fields change, what relationship exists between these changes. Since the circuit is ideal, there is no energy loss , so the total energy of the circuit remains constant).

With the start of recharging the capacitor, the discharge current will decrease to zero not immediately, but gradually. This is again due to the occurrence of counter-e. d.s. and inductive current of opposite direction. This current counteracts the decrease in the discharge current, as it previously counteracted its increase. Now it will support the main current. The energy of the magnetic field will decrease, the energy of the electric field will increase, the capacitor will be recharged.

Thus, the total energy of the oscillatory circuit at any time is equal to the sum of the energies of the magnetic and electric fields

The oscillations at which the energy of the electric field of the capacitor is periodically converted into the energy of the magnetic field of the coil are called ELECTROMAGNETIC oscillations. Since these fluctuations occur due to the initial supply of energy and without external influences, then they are FREE.

Lesson topic: Analogy between mechanical and electromagnetic oscillations.

Explanation of new material.

The purpose of the lesson: to explain the essence and prove the analogy between electromagnetic oscillations and oscillations of a spring pendulum using the dynamic oscillation model “Analogy between mechanical and electromagnetic oscillations” and PowerPoint presentations.

Material to repeat:

the concept of an oscillatory circuit;

the concept of an ideal oscillatory circuit;

conditions for the occurrence of fluctuations in c / c;

concepts of magnetic and electric fields;

fluctuations as a process of periodic energy change;

the energy of the circuit at an arbitrary point in time;

the concept of (free) electromagnetic oscillations.

(For repetition and consolidation, students are once again shown a dynamic model of an ideal oscillatory circuit).

In this lesson, we will look at the analogy between mechanical and electromagnetic oscillations. We will consider a spring pendulum as a mechanical oscillatory system.

(On the screen you see a dynamic model that demonstrates the analogy between mechanical and electromagnetic oscillations. It will help us understand oscillatory processes, both in a mechanical system and in an electromagnetic one).

So, in a spring pendulum, an elastically deformed spring imparts velocity to the load attached to it. A deformed spring has the potential energy of an elastically deformed body

a moving object has kinetic energy

The transformation of the potential energy of a spring into the kinetic energy of an oscillating body is a mechanical analogy of the transformation of the energy of the electric field of a capacitor into the energy of the magnetic field of a coil. In this case, the analog of the mechanical potential energy of the spring is the energy of the electric field of the capacitor, and the analog of the mechanical kinetic energy of the load is the energy of the magnetic field, which is associated with the movement of charges. Charging the capacitor from the battery corresponds to the message to the spring of potential energy (for example, displacement by hand).

Let's compare the formulas and derive general patterns for electromagnetic and mechanical vibrations.

From a comparison of the formulas, it follows that the analog of the inductance L is the mass m, and the analog of the displacement x is the charge q, the analog of the coefficient k is the reciprocal of the electrical capacity, i.e. 1/C.

The moment when the capacitor is discharged and the current strength reaches its maximum corresponds to the passage of the equilibrium position by the body at maximum speed (pay attention to the screens: you can observe this correspondence there).

As already mentioned in the last lesson, the movement of electrons along a conductor is conditional, because for them the main type of movement is oscillatory movement around the equilibrium position. Therefore, sometimes electromagnetic oscillations are compared with oscillations of water in communicating vessels (look at the screen, you can see that such an oscillatory system is located in the upper right corner), where each particle oscillates around the equilibrium position.

So, we found out that the analogy of inductance is mass, and the analogy of displacement is charge. But you know very well that a change in charge per unit of time is nothing more than a current strength, and a change in coordinates per unit of time is a speed, that is, q "= I, and x" = v. Thus, we have found another correspondence between mechanical and electrical quantities.

Let's make a table that will help us systematize the relationships between mechanical and electrical quantities in oscillatory processes.

Correspondence table between mechanical and electrical quantities in oscillatory processes.

Lesson topic: The equation of free harmonic oscillations in the circuit.

Explanation of new material.

The purpose of the lesson: the derivation of the basic equation of electromagnetic oscillations, the laws of change in charge and current strength, obtaining the Thomson formula and the expression for the natural frequency of the oscillation of the circuit using PowerPoint presentations.

Material to repeat:

the concept of electromagnetic oscillations;

the concept of the energy of an oscillatory circuit;

correspondence of electrical quantities to mechanical quantities during oscillatory processes.

(For repetition and consolidation, it is necessary to once again demonstrate the model of the analogy of mechanical and electromagnetic oscillations).

In the past lessons, we found out that electromagnetic oscillations, firstly, are free, and secondly, they represent a periodic change in the energies of the magnetic and electric fields. But in addition to energy, during electromagnetic oscillations, the charge also changes, and hence the current strength in the circuit and the voltage. In this lesson, we must find out the laws by which the charge changes, which means the current strength and voltage.

So, we found out that the total energy of the oscillatory circuit at any time is equal to the sum of the energies of the magnetic and electric fields: . We believe that the energy does not change with time, that is, the contour is ideal. This means that the time derivative of the total energy is equal to zero, therefore, the sum of the time derivatives of the energies of the magnetic and electric fields is equal to zero:

That is.

The minus sign in this expression means that when the energy of the magnetic field increases, the energy of the electric field decreases and vice versa. A physical meaning of this expression is such that the rate of change in the energy of the magnetic field is equal in absolute value and opposite in direction to the rate of change in the electric field.

Calculating the derivatives, we get

But, therefore, and - we got an equation describing free electromagnetic oscillations in the circuit. If we now replace q with x, x""=a x with q"", k with 1/C, m with L, we get the equation

describing the vibrations of a load on a spring. Thus, the equation of electromagnetic oscillations has the same mathematical form as the equation of oscillations of a spring pendulum.

As you saw in the demo model, the charge on the capacitor changes periodically. It is necessary to find the dependence of the charge on time.

From the ninth grade, you are familiar with the periodic functions sine and cosine. These functions have the following property: the second derivative of the sine and cosine is proportional to the functions themselves, taken with the opposite sign. Apart from these two, no other functions have this property. Now back to electric charge. We can safely say that the electric charge, and hence the current strength, during free oscillations change over time according to the law of cosine or sine, i.e. make harmonic vibrations. The spring pendulum also perform harmonic oscillations (acceleration is proportional to the displacement, taken with a minus sign).

So, in order to find the explicit dependence of the charge, current and voltage on time, it is necessary to solve the equation

taking into account the harmonic nature of the change in these quantities.

If we take an expression like q = q m cos t as a solution, then, when substituting this solution into the original equation, we get q""=-q m cos t=-q.

Therefore, as a solution, it is necessary to take an expression of the form

q=q m cossh o t,

where q m is the amplitude of charge oscillations (modulus the greatest value fluctuating value),

w o = - cyclic or circular frequency. Its physical meaning is

the number of oscillations in one period, i.e., for 2p s.

The period of electromagnetic oscillations is the period of time during which the current in the oscillatory circuit and the voltage on the capacitor plates make one complete oscillation. For harmonic oscillations T=2p s (smallest cosine period).

The oscillation frequency - the number of oscillations per unit time - is determined as follows: n = .

The frequency of free oscillations is called the natural frequency of the oscillatory system.

Since w o \u003d 2p n \u003d 2p / T, then T \u003d.

We defined the cyclic frequency as w o = , which means that for the period we can write

Т= = - Thomson's formula for the period of electromagnetic oscillations.

Then the expression for the natural oscillation frequency takes the form

It remains for us to obtain the equations for the oscillations of the current strength in the circuit and the voltage across the capacitor.

Since, then at q = q m cos u o t we get U=U m cos o t. This means that the voltage also changes according to the harmonic law. Let us now find the law according to which the current strength in the circuit changes.

By definition, but q=q m cosшt, so

where p/2 is the phase shift between current and charge (voltage). So, we found out that the current strength during electromagnetic oscillations also changes according to the harmonic law.

We considered an ideal oscillatory circuit in which there are no energy losses and free oscillations can continue indefinitely due to the energy once received from an external source. In a real circuit, part of the energy goes to heating the connecting wires and heating the coil. Therefore, free oscillations in the oscillatory circuit are damped.

Lesson topic.

Analogy between mechanical and electromagnetic oscillations.

Lesson Objectives:

Didactic – draw a complete analogy between mechanical and electromagnetic oscillations, revealing the similarities and differences between them;

educational – to show the universal nature of the theory of mechanical and electromagnetic oscillations;

Educational – develop the cognitive processes of students, based on the application scientific method knowledge: similarity and modeling;

Educational - to continue the formation of ideas about the relationship between natural phenomena and a single physical picture of the world, to teach to find and perceive beauty in nature, art and educational activities.

Type of lesson :

combined lesson

Work form:

individual, group

Methodological support :

computer, multimedia projector, screen, reference notes, self-study texts.

Intersubject communications :

physics

During the classes

Organizing time.

In today's lesson, we will draw an analogy between mechanical and electromagnetic oscillations.

II. Checking homework.

Physical dictation.

What is an oscillatory circuit made of?

The concept of (free) electromagnetic oscillations.

3. What needs to be done in order for electromagnetic oscillations to occur in the oscillatory circuit?

4. What device allows you to detect the presence of oscillations in the oscillatory circuit?

Knowledge update.

Guys, write down the topic of the lesson.

And now we will comparative characteristics two types of vibrations.

Front work with a class (checking is carried out through a projector).

(Slide 1)

Question for students: What do the definitions of mechanical and electromagnetic oscillations have in common and how do they differ!

General: in both types of oscillations, a periodic change in physical quantities occurs.

Difference: In mechanical vibrations - this is the coordinate, speed and acceleration In electromagnetic - charge, current and voltage.

(Slide 2)

Question for students: What do the methods of obtaining have in common and how do they differ?

General: both mechanical and electromagnetic oscillations can be obtained using oscillatory systems

Difference: various oscillatory systems - for mechanical ones - these are pendulums,and for electromagnetic - an oscillatory circuit.

(Slide3)

Question to students : "What do the demos shown have in common and how do they differ?"

General: the oscillatory system was removed from the equilibrium position and received a supply of energy.

Difference: the pendulums received a reserve of potential energy, and the oscillatory system received a reserve of energy of the electric field of the capacitor.

Question to students : Why electromagnetic oscillations cannot be observed as well as mechanical ones (visually)

Answer: since we cannot see how the capacitor is charging and recharging, how the current flows in the circuit and in what direction, how the voltage between the capacitor plates changes

Independent work

(Slide3)

Students are asked to complete the table on their own.Correspondence between mechanical and electrical quantities in oscillatory processes

III. Fixing the material

Reinforcing test on this topic:

1. The period of free oscillations of a thread pendulum depends on...
A. From the mass of the cargo. B. From the length of the thread. B. From the frequency of oscillations.

2. The maximum deviation of the body from the equilibrium position is called ...
A. Amplitude. B. Offset. During the period.

3. The oscillation period is 2 ms. The frequency of these oscillations isA. 0.5 Hz B. 20 Hz C. 500 Hz

(Answer:Given:
mswith Find:
Solution: Hz
Answer: 20 Hz)

4. Oscillation frequency 2 kHz. The period of these oscillations is
A. 0.5 s B. 500 µs C. 2 s(Answer:T= 1\n= 1\2000Hz = 0.0005)

5. The oscillatory circuit capacitor is charged so that the charge on one of the capacitor plates is + q. After what is the minimum time after the capacitor is closed to the coil, the charge on the same capacitor plate becomes equal to - q, if the period of free oscillations in the circuit is T?
A. T/2 B. T V. T/4

(Answer:A) Т/2because even after T/2 the charge becomes +q again)

6. How many complete oscillations does material point for 5 s if the oscillation frequency is 440 Hz?
A. 2200 B. 220 V. 88

(Answer:U=n\t hence n=U*t ; n=5 s * 440 Hz=2200 vibrations)

7. In an oscillatory circuit consisting of a coil, a capacitor and a key, the capacitor is charged, the key is open. After what time after the switch is closed, the current in the coil will increase to a maximum value if the period of free oscillations in the circuit is equal to T?
A. T/4 B. T/2 W. T

(Answer:Answer T/4at t=0 the capacitance is charged, the current is zerothrough T / 4 the capacity is discharged, the current is maximumthrough T / 2, the capacitance is charged with the opposite voltage, the current is zerothrough 3T / 4 the capacity is discharged, the current is maximum, opposite to that at T / 4through T the capacitance is charged, the current is zero (the process is repeated)

8. The oscillatory circuit consists
A. Capacitor and resistor B. Capacitor and bulb C. Capacitor and inductor

IV . Homework

G. Ya. Myakishev§18, pp.77-79

Answer the questions:

1. In what system do electromagnetic oscillations occur?

2. How is the transformation of energies carried out in the circuit?

3. Write down the energy formula at any time.

4. Explain the analogy between mechanical and electromagnetic oscillations.

V . Reflection

Today I found out...

it was interesting to know...

it was hard to do...

now I can decide..

I have learned (learned)...

I managed…

I could)…

I will try myself...

(Slide1)

(Slide2)

(Slide3)

(Slide 4)

Electrical and magnetic phenomena are inextricably linked. A change in the electrical characteristics of a phenomenon entails a change in its magnetic characteristics. Electromagnetic oscillations are of particular practical value.

Electromagnetic vibrations- these are interrelated changes in the electric and magnetic fields, in which the values of the quantities characterizing the system (electric charge, current, voltage, energy) are repeated to one degree or another.

It should be noted that between fluctuations of different physical nature there is an analogy. They are described by the same differential equations and functions. Therefore, the information obtained in the study of mechanical oscillations is also useful in the study of electromagnetic oscillations.

In modern technology, electromagnetic oscillations and waves play a greater role than mechanical ones, as they are used in communication devices, television, radar, and in various technological processes that have determined scientific and technological progress.

Electromagnetic oscillations are excited in an oscillatory system called oscillatory circuit. It is known that any conductor has electrical resistance R, electric capacity WITH and inductance L, and these parameters are dispersed along the length of the conductor. Lumped parameters R, WITH, L possess a resistor, a capacitor and a coil, respectively.

An oscillatory circuit is a closed electrical circuit consisting of a resistor, a capacitor and a coil (Fig. 4.1). Such a system is similar to a mechanical pendulum.

The circuit is in a state of equilibrium if there are no charges and currents in it. To bring the circuit out of balance, it is necessary to charge the capacitor (or to excite an induction current with the help of a changing magnetic field). Then an electric field with intensity will appear in the capacitor. When the key is closed TO current will flow in the circuit, as a result, the capacitor will be discharged, the energy of the electric field will decrease, and the energy of the magnetic field of the inductor will increase.

Rice. 4.1 Oscillatory circuit

At some point in time, equal to a quarter of the period, the capacitor is completely discharged, and the magnetic field reaches its maximum. This means that the energy of the electric field has been converted into the energy of a magnetic field. Since the currents supporting the magnetic field have disappeared, it will begin to decrease. The decreasing magnetic field causes a self-induction current, which, according to Lenz's law, is directed in the same way as the discharge current. Therefore, the capacitor will be recharged and an electric field will appear between its plates with a strength opposite to the original one. After a time equal to half the period, the magnetic field will disappear, and the electric field will reach a maximum.

Then all processes will occur in the opposite direction and after a time equal to the oscillation period, the oscillatory circuit will return to its original state with a capacitor charge. Consequently, electrical oscillations occur in the circuit.

For a complete mathematical description of the processes in the circuit, it is necessary to find the law of change in one of the quantities (for example, charge) over time, which, using the laws of electromagnetism, will allow us to find the patterns of change in all other quantities. The functions describing the change in the quantities characterizing the processes in the circuit are the solution of the differential equation. Ohm's law and Kirchhoff's rules are used to compile it. However, they are performed for direct current.

An analysis of the processes occurring in an oscillatory circuit showed that the laws of direct current can also be applied to a time-varying current that satisfies the condition of quasi-stationarity. This condition consists in the fact that during the propagation of the disturbance to the most remote point of the circuit, the current strength and voltage change insignificantly, then the instantaneous values of the electrical quantities at all points of the circuit are practically the same. Since the electromagnetic field propagates in a conductor at the speed of light in vacuum, the propagation time of perturbations is always less than the period of current and voltage oscillations.

In the absence of an external source in the oscillatory circuit, free electromagnetic vibrations.

According to the second rule of Kirchhoff, the sum of the voltages across the resistor and across the capacitor is equal to the electromotive force, in this case, the self-induction EMF that occurs in the coil when a changing current flows in it

Taking into account that , and, therefore, , we represent the expression (4.1) in the form:

. (4.2)