Free electromagnetic oscillations. Conversion of energy in an oscillatory circuit. Thompson's formula. Oscillatory circuit. Thomson's formula Derivation of Thomson's formula

Occupation type: a lesson of initial acquaintance with the material and the practical application of knowledge and skills.

Duration of the lesson: 45 minutes.

Goals:

Didactic - to generalize and systematize knowledge about the physical processes occurring in the electromagnetic oscillatory circuit

create conditions for the assimilation of new material using active teaching methods

Educational I am- to show the universal nature of the theory of oscillations;

Developing - to develop the cognitive processes of students, based on the application of the scientific method of cognition: analogy and modeling; forecasting the situation; to develop among schoolchildren the methods of effective processing of educational information, to continue the formation of communicative competencies.

Educational - to continue the formation of ideas about the relationship between natural phenomena and a single physical picture of the world

Lesson Objectives:

1... Educational

ü to formulate the dependence of the period of the oscillating circuit on its characteristics: capacitance and inductance

ü to study the techniques of solving typical problems on the "Oscillatory circuit"

2. Developing

ü continue the formation of skills to compare phenomena, draw conclusions and generalizations based on the experiment

ü work on the formation of skills to analyze properties and phenomena based on knowledge.

3. Educational

ü to show the importance of experimental facts and experiment in human life.

ü to reveal the meaning of the accumulation of facts and their clarifications in the cognition of phenomena.

ü to acquaint students with the relationship and conditionality of the phenomena of the surrounding world.

TSO:computer, projector, IAD

Preliminary preparation:

- individual assessment sheets - 24pcs

- route sheets (colored) - 4pcs

Technological lesson map:

• Electromagnetic vibrations Are periodic changes over time in electrical and magnetic quantities in an electrical circuit.

• Free are called such hesitation that arise in a closed system due to the deviation of this system from the state of stable equilibrium.

During oscillations, there is a continuous process of converting the energy of the system from one form to another. In the case of fluctuations in the electromagnetic field, exchange can only take place between the electric and magnetic components of this field. The simplest system where this process can take place is oscillatory circuit.

• Ideal oscillating circuit (LC circuit) - an electrical circuit consisting of a coil of inductance L and a capacitor with a capacity C.

Unlike a real oscillatory circuit, which has an electrical resistance R, the electrical resistance of an ideal circuit is always zero. Therefore, an ideal oscillating circuit is a simplified model of a real circuit.

Figure 1 shows a diagram of an ideal oscillatory circuit.

Circuit energies

Total energy of the oscillating circuit

\ (W = W_ (e) + W_ (m), \; \; \; W_ (e) = \ dfrac (C \ cdot u ^ (2)) (2) = \ dfrac (q ^ (2)) (2C), \; \; \; W_ (m) = \ dfrac (L \ cdot i ^ (2)) (2), \)

Where W e- the energy of the electric field of the oscillating circuit at a given time, WITH- electrical capacity of the capacitor, u- the value of the voltage across the capacitor at a given time, q- the value of the capacitor charge at a given time, W m- the energy of the magnetic field of the oscillating circuit at a given time, L- coil inductance, i- the value of the current in the coil at a given time.

Processes in an oscillatory circuit

Consider the processes that occur in the oscillatory circuit.

To remove the circuit from the equilibrium position, charge the capacitor so that there will be a charge on its plates Q m(fig. 2, position 1 ). Taking into account the equation \ (U_ (m) = \ dfrac (Q_ (m)) (C) \) we find the value of the voltage across the capacitor. There is no current in the circuit at this moment in time, i.e. i = 0.

After the key is closed, under the action of the electric field of the capacitor, an electric current appears in the circuit, the current strength i which will increase over time. The capacitor will start to discharge at this time, because the electrons that create the current (I remind you that the direction of movement of positive charges is taken as the direction of the current) leave the negative plate of the capacitor and come to the positive one (see Fig. 2, position 2 ). Together with the charge q the tension will also decrease u\ (\ left (u = \ dfrac (q) (C) \ right). \) With an increase in the current through the coil, an EMF of self-induction will appear, which prevents the change in current. As a result, the current in the oscillatory circuit will increase from zero to a certain maximum value not instantly, but within a certain period of time, determined by the inductance of the coil.

Capacitor charge q decreases and at some point in time becomes equal to zero ( q = 0, u= 0), the current in the coil will reach a certain value I m(see fig. 2, position 3 ).

Without the capacitor's electric field (and resistance), the electrons that create the current continue to move by inertia. In this case, the electrons arriving at the neutral plate of the capacitor impart a negative charge to it, the electrons leaving the neutral plate impart a positive charge to it. A charge begins to appear on the capacitor q(and voltage u), but of the opposite sign, i.e. the capacitor is being recharged. Now the new electric field of the capacitor prevents the movement of electrons, so the current strength i begins to decrease (see Fig. 2, position 4 ). Again, this does not happen instantaneously, since now the EMF of self-induction seeks to compensate for the decrease in current and "maintains" it. And the value of the current I m(pregnant 3 ) turns out maximum current strength in the contour.

And again, under the action of the electric field of the capacitor, an electric current will appear in the circuit, but directed in the opposite direction, the current strength i which will increase over time. And the capacitor will be discharged at this time (see Fig. 2, position 6 ) to zero (see Fig. 2, position 7 ). Etc.

Since the charge on the capacitor q(and voltage u) determines its electric field energy W e\ (\ left (W_ (e) = \ dfrac (q ^ (2)) (2C) = \ dfrac (C \ cdot u ^ (2)) (2) \ right), \) and the current in the coil i- magnetic field energy Wm\ (\ left (W_ (m) = \ dfrac (L \ cdot i ^ (2)) (2) \ right), \) then along with changes in charge, voltage and current, energies will also change.

Designations in the table:

\ (W_ (e \, \ max) = \ dfrac (Q_ (m) ^ (2)) (2C) = \ dfrac (C \ cdot U_ (m) ^ (2)) (2), \; \; \; W_ (e \, 2) = \ dfrac (q_ (2) ^ (2)) (2C) = \ dfrac (C \ cdot u_ (2) ^ (2)) (2), \; \; \ ; W_ (e \, 4) = \ dfrac (q_ (4) ^ (2)) (2C) = \ dfrac (C \ cdot u_ (4) ^ (2)) (2), \; \; \; W_ (e \, 6) = \ dfrac (q_ (6) ^ (2)) (2C) = \ dfrac (C \ cdot u_ (6) ^ (2)) (2), \)

\ (W_ (m \; \ max) = \ dfrac (L \ cdot I_ (m) ^ (2)) (2), \; \; \; W_ (m2) = \ dfrac (L \ cdot i_ (2 ) ^ (2)) (2), \; \; \; W_ (m4) = \ dfrac (L \ cdot i_ (4) ^ (2)) (2), \; \; \; W_ (m6) = \ dfrac (L \ cdot i_ (6) ^ (2)) (2). \)

The total energy of an ideal oscillating circuit is conserved over time, since it contains energy losses (no resistance). Then

\ (W = W_ (e \, \ max) = W_ (m \, \ max) = W_ (e2) + W_ (m2) = W_ (e4) + W_ (m4) = ... \)

Thus, ideally LC-the loop will experience periodic changes in the current strength i, charge q and voltage u, and the total energy of the circuit will remain constant. In this case, they say that free electromagnetic oscillations.

• Free electromagnetic oscillations in the circuit - these are periodic changes in the charge on the capacitor plates, current strength and voltage in the circuit, occurring without the consumption of energy from external sources.

Thus, the emergence of free electromagnetic oscillations in the circuit is due to the overcharging of the capacitor and the emergence of an EMF of self-induction in the coil, which "provides" this overcharge. Note that the charge of the capacitor q and the current in the coil i reach their maximum values Q m and I m at different times.

Free electromagnetic oscillations in the circuit occur according to the harmonic law:

\ (q = Q_ (m) \ cdot \ cos \ left (\ omega \ cdot t + \ varphi _ (1) \ right), \; \; \; u = U_ (m) \ cdot \ cos \ left (\ omega \ cdot t + \ varphi _ (1) \ right), \; \; \; i = I_ (m) \ cdot \ cos \ left (\ omega \ cdot t + \ varphi _ (2) \ right). \)

The smallest period of time during which LC- the circuit returns to its original state (to the initial value of the charge of this plate), called the period of free (natural) electromagnetic oscillations in the circuit.

The period of free electromagnetic oscillations in LC-the contour is determined by the Thomson formula:

\ (T = 2 \ pi \ cdot \ sqrt (L \ cdot C), \; \; \; \ omega = \ dfrac (1) (\ sqrt (L \ cdot C)). \)

From the point of view of a mechanical analogy, an ideal oscillatory circuit corresponds to a spring pendulum without friction, and a real one with friction. Due to the action of frictional forces, the oscillations of the spring pendulum damp over time.

* Derivation of Thomson's formula

Since the full energy of the ideal LC- the contour equal to the sum of the energies of the electrostatic field of the capacitor and the magnetic field of the coil is preserved, then at any moment of time the equality

\ (W = \ dfrac (Q_ (m) ^ (2)) (2C) = \ dfrac (L \ cdot I_ (m) ^ (2)) (2) = \ dfrac (q ^ (2)) (2C ) + \ dfrac (L \ cdot i ^ (2)) (2) = (\ rm const). \)

We obtain the equation of oscillations in LC-contour using the law of conservation of energy. Differentiating the expression for its total energy in time, taking into account that

\ (W "= 0, \; \; \; q" = i, \; \; \; i "= q" ", \)

we obtain an equation describing free vibrations in an ideal contour:

\ (\ left (\ dfrac (q ^ (2)) (2C) + \ dfrac (L \ cdot i ^ (2)) (2) \ right) ^ ((")) = \ dfrac (q) (C ) \ cdot q "+ L \ cdot i \ cdot i" = \ dfrac (q) (C) \ cdot q "+ L \ cdot q" \ cdot q "" = 0, \)

\ (\ dfrac (q) (C) + L \ cdot q "" = 0, \; \; \; \; q "" + \ dfrac (1) (L \ cdot C) \ cdot q = 0. \ )

Rewriting it as:

\ (q "" + \ omega ^ (2) \ cdot q = 0, \)

we note that this is the equation of harmonic oscillations with a cyclic frequency

\ (\ omega = \ dfrac (1) (\ sqrt (L \ cdot C)). \)

Accordingly, the period of the considered fluctuations

\ (T = \ dfrac (2 \ pi) (\ omega) = 2 \ pi \ cdot \ sqrt (L \ cdot C). \)

Literature

1. Zhilko, V.V. Physics: textbook. general education allowance for grade 11. shk. from rus. lang. training / V.V. Zhilko, L.G. Markovich. - Minsk: Nar. Asveta, 2009 .-- S. 39-43.

Lesson number 48-169 Oscillatory circuit. Free electromagnetic oscillations. Conversion of energy in an oscillatory circuit. Thompson's formula.Fluctuations- movements or conditions that are repeated in time.Electromagnetic vibrations -these are vibrations of electrical andmagnetic fields that resistdriven by periodic infidelitycharge, current and voltage. An oscillating circuit is a system consisting of an inductor and a capacitor(fig. a). If the capacitor is charged and shorted to the coil, then a current will flow through the coil (Fig. B). When the capacitor is discharged, the current in the circuit will not stop due to self-induction in the coil. The induction current, according to Lenz's rule, will flow in the same direction and recharge the capacitor (Fig. C). The current in this direction will stop, and the process will be repeated in the opposite direction (Fig. G).

Thus, in vibrationthe original circuit of theelectromagnetic vibrationsdue to energy conversionelectric field condensatera( W E =
) into the energy of the magnetic field of the coil with current(W М =
), and vice versa.

Harmonic oscillations - periodic changes in a physical quantity depending on time, occurring according to the law of sine or cosine.

The equation describing free electromagnetic oscillations takes the form

q "= - ω 0 2 q (q" is the second derivative.

The main characteristics of the oscillatory motion:

The oscillation period is the minimum period of time T, after which the process is completely repeated.

The amplitude of harmonic oscillations is the modulus of the largest value of the oscillating quantity.

Knowing the period, it is possible to determine the frequency of oscillations, i.e., the number of oscillations per unit of time, for example, per second. If one oscillation occurs in time T, then the number of oscillations in 1 s ν is determined as follows: ν = 1 / T.

Recall that in the International System of Units (SI), the vibration frequency is equal to one if one vibration occurs in 1 s. The unit of frequency is called hertz (abbreviated: Hz) after the German physicist Heinrich Hertz.

After a period of time equal to the period T, i.e., as the argument of the cosine increases by ω 0 T, the charge value is repeated and the cosine returns to the previous value. It is known from the mathematics course that the smallest period of the cosine is 2n. Therefore, ω 0 T= 2π, whence ω 0 = = 2πν Thus, the quantity ω 0 - this is the number of vibrations, but not for 1 s, but for 2 yrs. It is called cyclical or circular frequency.

The frequency of free vibrations is called natural oscillatory frequencysystems. In what follows, we will often refer to the cyclic frequency simply as frequency for brevity. Distinguish the cyclic frequency ω 0 on the frequency ν can be by designations.

By analogy with the solution of the differential equation for a mechanical oscillatory system cyclic frequency free electricallyhesitation is equal to: ω 0 =

The period of free oscillations in the circuit is equal to: T = = 2π
- Thomson's formula.

The phase of oscillations (from the Greek word phasis - appearance, stage of development of any phenomenon) is the value of φ, standing under the sign of cosine or sine. The phase is expressed in angular units - radians. The phase determines, at a given amplitude, the state of the oscillatory system at any time.

Oscillations with the same amplitudes and frequencies can differ from each other in phases.

Since ω 0 =, then φ = ω 0 T = 2π... The ratio shows how much of the period has passed since the beginning of the oscillations. Any time value expressed in fractions of a period corresponds to a phase value expressed in radians. So, after time t = (quarter period) φ = , after half the period φ = π, after the whole period φ = 2π, etc. You can plot the dependence

charge not from time, but from phase. The figure shows the same cosine as in the previous one, but the horizontal axis is plotted instead of time

different values ​​of the phase φ.

Correspondence between mechanical and electrical quantities in oscillatory processes

Mechanical quantities

942(932). The initial charge imparted to the capacitor of the oscillatory circuit was reduced by 2 times. How many times have changed: a) voltage amplitude; b) the amplitude of the current strength;

c) the total energy of the electric field of the capacitor and the magnetic field of the coil?

943(933). With an increase in the voltage across the capacitor of the oscillatory circuit by 20 V, the amplitude of the current increased by 2 times. Find the initial voltage.

945(935). The oscillating circuit consists of a capacitor with a capacity of C = 400 pF and a coil with inductance L = 10 mH. Find the amplitude of the current I T , if the amplitude of voltage fluctuations U T = 500 V.

952(942). After what time (in fractions of the period t / T) on the capacitor of the oscillatory circuit for the first time there will be a charge equal to half the amplitude value?

957(947). What kind of inductor should be included in the oscillatory circuit in order to obtain a free oscillation frequency of 10 MHz with a capacitor of 50 pF?

Oscillatory circuit. Period of free fluctuations.

1. After the charge has been imparted to the capacitor of the oscillatory circuit q = 10 -5 C, damped oscillations appeared in the circuit. How much heat will be released in the circuit by the time the oscillations in it are completely damped? The capacity of the capacitor is C = 0.01μF.

2. The oscillating circuit consists of a 400nF capacitor and a 9μH inductance coil. What is the period of natural oscillations of the circuit?

3. What inductance should be included in the oscillatory circuit in order to obtain a period of natural oscillations of 2 ∙ 10 -6 s with a capacitance of 100 pF.

4. Compare the stiffness of the springs k1 / k2 of two pendulums with weights of 200g and 400g, respectively, if the periods of their oscillations are equal.

5. Under the action of a motionlessly hanging weight on the spring, its elongation was 6.4 cm. Then the load was pulled back and released, as a result of which it began to hesitate. Determine the period of these fluctuations.

6. A weight was suspended from the spring, removed from its equilibrium position and released. The load began to oscillate with a period of 0.5 s. Determine the elongation of the spring after the oscillation stops. Disregard the weight of the spring.

Exam questions:

1. Which of the following expressions determines the period of free oscillations in the oscillatory circuit? A.; B.
; V.
; G.
; D. 2.

2... Which of the following expressions determines the cyclic frequency of free oscillations in an oscillatory circuit? A. B.
V.
G.
D. 2π

3. The figure shows a graph of the dependence of the X coordinate of a body performing harmonic oscillations along the ox axis on time. What is the period of oscillation of the body?

4. The figure shows the profile of the wave at a certain moment in time. What is its length?

5... The figure shows a graph of the dependence of the current through the coil of the oscillating circuit on time. What is the period of current fluctuations? A. 0.4 s. B. 0.3 s. B. 0.2 s. D. 0.1 s.

E. There is no correct answer among the answers A-D.

A. 0.2 m. B. 0.4 m. C. 4 m. D. 8 m. D. 12 m.

7. Electrical oscillations in the oscillatory circuit are given by the equation q = 10 -2 ∙ cos 20t (Cl).

What is the amplitude of charge oscillations?

A . 10 -2 Cl. B.cos 20t Cl. B.20t Cl. D.20 Cl. D. Among answers A-D there is no correct one.

8. With harmonic vibrations along the OX axis, the coordinate of the body changes according to the law X = 0.2cos (5t + ). What is the amplitude of body vibrations?

A. Hmm; B. 0.2 m; B. cos (5t +) m; (5t +) m; D. m

9. The oscillation frequency of the wave source is 0.2 s -1, the wave propagation speed is 10 m / s. What is the wavelength? A. 0.02 m. B. 2 m. C. 50 m.

D. By the condition of the problem, it is impossible to determine the wavelength. E. There is no correct answer among the answers A-D.

10. Wavelength 40 m, propagation speed 20 m / s. What is the oscillation frequency of the wave source?

A. 0.5 s -1. B. 2 s -1. H. 800 s -1.

D. By the condition of the problem, it is impossible to determine the oscillation frequency of the wave source.

E. There is no correct answer among the answers A-D.

3

"Damped oscillations" - 26.1. Free damped mechanical vibrations; 26.2. Damping factor and logarithmic damping decrement; 26.26. Self-oscillations; Today: Saturday, August 6, 2011 Lecture 26. Fig. 26.1.

"Harmonic vibrations" - The beat method is used for tuning musical instruments, hearing analysis, and more. Figure 4. Oscillations of the species. (2.2.4). ? 1 - phase of the 1st oscillation. - The resulting vibration, also harmonic, with a frequency ?: The projection of a circular motion onto the y-axis also makes a harmonic vibration. Figure 3.

"Oscillation frequency" - Sound reflection. Sound speed in various media, m / s (at t = 20 ° C). Mechanical vibrations with a frequency of less than 20Hz are called infrasound. Parse sound as a phenomenon. Objectives of the project. Sound sources. The speed of sound depends on the properties of the medium in which the sound travels. What determines the timbre of a sound?

"Mechanical vibrations and waves" - Properties of waves. Types of waves. Mathematical pendulum. The period of free oscillations of a mathematical pendulum. Conversion of energy. Reflection laws. Spring pendulum. The hearing organs are most sensitive to sounds with frequencies from 700 to 6000 Hz. Free Forced Self-Oscillations.

"Mechanical vibrations" - Harmonic. Elastic waves are mechanical disturbances propagating in an elastic medium. Mathematical pendulum. Waves. Wavelength (?) - the distance between the nearest particles, vibrating in the same phase. Forced. Forced vibrations. The graph of a mathematical pendulum. Waves are the propagation of vibrations in space over time.

"Mechanical resonance" - Amplitude of forced vibrations. State educational institution Gymnasium No. 363 of the Frunzensky District. The destructive role of resonance Bridges. Resonance in technology. Thomas Jung. 1. Physical foundations of resonance Forced oscillations. A mechanical reed frequency meter is a device for measuring vibration frequency.

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