The maximum current strength formula in the oscillatory circuit. Oscillatory circuit

An electromagnetic field can also exist in the absence of electric charges or currents: it is precisely such “self-sustaining” electric and magnetic fields that represent electromagnetic waves which include visible light, infrared, ultraviolet and x-ray radiation, radio waves, etc.

§ 25. Oscillatory circuit

The simplest system in which natural electromagnetic oscillations are possible is the so-called oscillatory circuit, consisting of a capacitor and an inductor connected to each other (Fig. 157). Like a mechanical oscillator, such as a massive body on an elastic spring, natural oscillations in the circuit are accompanied by energy transformations.

Rice. 157. Oscillatory circuit

Analogy between mechanical and electromagnetic oscillations. For an oscillatory circuit, the analogue of the potential energy of a mechanical oscillator (for example, the elastic energy of a deformed spring) is the energy of the electric field in a capacitor. An analogue of the kinetic energy of a moving body is the energy magnetic field in the inductor. Indeed, the energy of the spring is proportional to the square of the displacement from the equilibrium position, and the energy of the capacitor is proportional to the square of the charge. The kinetic energy of the body is proportional to the square of its velocity, and the energy of the magnetic field in the coil is proportional to the square of the current.

The total mechanical energy of the spring oscillator E is equal to the sum of the potential and kinetic energies:

Vibration energy. Similarly, the total electromagnetic energy of an oscillatory circuit is equal to the sum of the energies of the electric field in the capacitor and the magnetic field in the coil:

From a comparison of formulas (1) and (2) it follows that the analogue of the stiffness k of the spring oscillator in the oscillatory circuit is the reciprocal value of the capacitance C, and the analogue of the mass is the inductance of the coil

Recall that in a mechanical system, the energy of which is given by expression (1), own undamped harmonic oscillations can occur. The square of the frequency of such oscillations is equal to the ratio of the coefficients at the squares of displacement and velocity in the expression for energy:

Own frequency. In an oscillatory circuit, the electromagnetic energy of which is given by expression (2), own undamped harmonic oscillations can occur, the square of the frequency of which is also, obviously, equal to the ratio of the corresponding coefficients (i.e., the coefficients at the squares of charge and current strength):

From (4) follows the expression for the oscillation period, called the Thomson formula:

With mechanical oscillations, the dependence of the displacement x on time is determined by a cosine function, the argument of which is called the oscillation phase:

Amplitude and initial phase. The amplitude A and the initial phase a are determined by the initial conditions, i.e., the values of the displacement and velocity at

Similarly, with electromagnetic natural oscillations in the circuit, the charge of the capacitor depends on time according to the law

where the frequency is determined, in accordance with (4), only by the properties of the circuit itself, and the amplitude of the charge oscillations and the initial phase a, as in the case of a mechanical oscillator, are determined

initial conditions, i.e., the values of the charge of the capacitor and the current strength at Thus, the natural frequency does not depend on the method of excitation of oscillations, while the amplitude and initial phase are determined precisely by the conditions of excitation.

Energy transformations. Let us consider in more detail the energy transformations during mechanical and electromagnetic oscillations. On fig. 158 schematically shows the states of the mechanical and electromagnetic oscillators at time intervals of a quarter of a period

Rice. 158. Energy transformations during mechanical and electromagnetic vibrations

Twice during the period of oscillation, energy is converted from one form to another and vice versa. The total energy of the oscillatory circuit, like the total energy of the mechanical oscillator, remains unchanged in the absence of dissipation. To verify this, it is necessary to substitute the expression (6) for and the expression for the current strength into formula (2)

Using formula (4) for we obtain

Rice. 159. Graphs of the energy of the electric field of the capacitor and the energy of the magnetic field in the coil as a function of the charge time of the capacitor

The constant total energy coincides with the potential energy at the moments when the charge of the capacitor is maximum, and coincides with the energy of the magnetic field of the coil - "kinetic" energy - at the moments when the charge of the capacitor vanishes and the current is at its maximum. During mutual transformations, two types of energy make harmonic oscillations with the same amplitude in antiphase with each other and with a frequency relative to their average value. This is easy to verify as from Fig. 158, and with the help of formulas trigonometric functions half argument:

Graphs of the dependence of the energy of the electric field and the energy of the magnetic field on the charge time of the capacitor are shown in fig. 159 for the initial phase

The quantitative regularities of natural electromagnetic oscillations can be established directly on the basis of the laws for quasi-stationary currents, without resorting to the analogy with mechanical oscillations.

The equation for oscillations in the circuit. Consider the simplest oscillatory circuit shown in Fig. 157. When bypassing the circuit, for example, counterclockwise, the sum of the voltages on the inductor and capacitor in such a closed series circuit is zero:

The voltage on the capacitor is related to the charge of the plate and to the capacitance With the relation The voltage on the inductance at any time is equal in absolute value and opposite in sign EMF self-induction, so the current in the circuit is equal to the rate of change of the charge of the capacitor:

We get Now expression (10) takes the form

Let's rewrite this equation differently, introducing by definition:

Equation (12) coincides with the equation harmonic vibrations mechanical oscillator with natural frequency The solution of such an equation is given by the harmonic (sinusoidal) function of time (6) with arbitrary values of the amplitude and initial phase a. From this follow all the above results concerning electromagnetic oscillations in the circuit.

Attenuation of electromagnetic oscillations. So far, we have discussed eigenoscillations in an idealized mechanical system and an idealized LC circuit. The idealization was to neglect the friction in the oscillator and the electrical resistance in the circuit. Only in this case the system will be conservative and the energy of oscillations will be conserved.

Rice. 160. Oscillatory circuit with resistance

Accounting for the dissipation of the energy of oscillations in the circuit can be carried out in the same way as it was done in the case of a mechanical oscillator with friction. The presence of electrical resistance of the coil and connecting wires is inevitably associated with the release of Joule heat. As before, this resistance can be viewed as independent element in wiring diagram oscillatory circuit, considering the coil and wires to be ideal (Fig. 160). When considering a quasi-stationary current in such a circuit, in equation (10) it is necessary to add the voltage across the resistance

Substituting into we get

Introducing the notation

we rewrite equation (14) in the form

Equation (16) for has exactly the same form as the equation for for vibrations of a mechanical oscillator with

friction proportional to speed (viscous friction). Therefore, in the presence of electrical resistance in the circuit, electromagnetic oscillations occur according to the same law as the mechanical oscillations of an oscillator with viscous friction.

Dissipation of vibration energy. As with mechanical vibrations, it is possible to establish the law of decrease in the energy of natural vibrations with time, applying the Joule-Lenz law to calculate the released heat:

As a result, in the case of low damping for time intervals much longer than the period of oscillations, the rate of decrease in the energy of oscillations turns out to be proportional to the energy itself:

The solution of equation (18) has the form

The energy of natural electromagnetic oscillations in a circuit with resistance decreases exponentially.

The energy of oscillations is proportional to the square of their amplitude. For electromagnetic oscillations, this follows, for example, from (8). Therefore, the amplitude of damped oscillations, in accordance with (19), decreases according to the law

Lifetime of oscillations. As can be seen from (20), the amplitude of the oscillations decreases by a factor of 1 in a time equal to, regardless of the initial value of the amplitude. This time x is called the lifetime of the oscillations, although, as can be seen from (20), the oscillations formally continue indefinitely. In reality, of course, it makes sense to talk about oscillations only as long as their amplitude exceeds the characteristic value of the thermal noise level in a given circuit. Therefore, in fact, the oscillations in the circuit "live" for a finite time, which, however, can be several times greater than the lifetime x introduced above.

It is often important to know not the lifetime of oscillations x itself, but the number of complete oscillations that will occur in the circuit during this time x. This number multiplied by is called the quality factor of the circuit.

Strictly speaking, damped oscillations are not periodic. With a small attenuation, we can conditionally speak of a period, which is understood as the time interval between two

successive maximum values of the charge of the capacitor (of the same polarity), or maximum values of the current (of one direction).

The damping of the oscillations affects the period, leading to its increase in comparison with the idealized case of no damping. With low damping, the increase in the oscillation period is very small. However, with strong attenuation, there may be no oscillations at all: a charged capacitor will discharge aperiodically, i.e., without changing the direction of the current in the circuit. So it will be with i.e. with

Exact solution. The patterns of damped oscillations formulated above follow from the exact solution of differential equation (16). By direct substitution, one can verify that it has the form

where are arbitrary constants whose values are determined from the initial conditions. For low damping, the cosine multiplier can be viewed as a slowly varying oscillation amplitude.

Task

Recharging capacitors through an inductor. In the circuit, the diagram of which is shown in Fig. 161, the charge of the upper capacitor is equal and the lower one is not charged. At the moment the key is closed. Find the time dependence of the charge of the upper capacitor and the current in the coil.

Rice. 161. Only one capacitor is charged at the initial moment of time

Rice. 162. Charges of capacitors and current in the circuit after closing the key

Rice. 163. Mechanical analogy for the electrical circuit shown in fig. 162

Decision. After the key is closed, oscillations occur in the circuit: the upper capacitor begins to discharge through the coil, while charging the lower one; then everything happens in the opposite direction. Let, for example, at , the upper plate of the capacitor be positively charged. Then

after a short period of time, the signs of the charges of the capacitor plates and the direction of the current will be as shown in Fig. 162. Denote by the charges of those plates of the upper and lower capacitors, which are interconnected through an inductor. Based on the conservation law electric charge

The sum of stresses on all elements of a closed circuit at each moment of time is equal to zero:

The sign of the voltage on the capacitor corresponds to the distribution of charges in fig. 162. and the indicated direction of current. The expression for the current through the coil can be written in either of two forms:

Let us exclude from the equation using relations (22) and (24):

Introducing the notation

we rewrite (25) in the following form:

If instead of introducing the function

and take into account that (27) takes the form

This is the usual equation of undamped harmonic oscillations, which has a solution

where and are arbitrary constants.

Returning from the function, we obtain the following expression for the dependence on the charge time of the upper capacitor:

To determine the constants and a, we take into account that at the initial moment the charge a current For the current strength from (24) and (31) we have

Since it follows from here that Substituting now in and taking into account that we get

So, the expressions for charge and current strength are

The nature of the charge and current oscillations is especially evident when the same values capacitor capacities. In this case

The charge of the upper capacitor oscillates with an amplitude of about an average value equal to Half the oscillation period, it decreases from the maximum value at the initial moment to zero, when the entire charge is on the lower capacitor.

Expression (26) for the oscillation frequency, of course, could be written immediately, since in the circuit under consideration the capacitors are connected in series. However, it is difficult to write expressions (34) directly, since under such initial conditions it is impossible to replace the capacitors included in the circuit with one equivalent one.

A visual representation of the processes taking place here is given by the mechanical analogue of this electrical circuit, shown in Fig. 163. Identical springs correspond to the case of condensers of the same capacity. At the initial moment, the left spring is compressed, which corresponds to a charged capacitor, and the right one is in an undeformed state, since the degree of deformation of the spring serves as an analogue of the capacitor charge. When passing through the middle position, both springs are partially compressed, and in the extreme right position, the left spring is not deformed, and the right one is compressed in the same way as the left one at the initial moment, which corresponds to the complete flow of charge from one capacitor to another. Although the ball performs the usual harmonic oscillations around the equilibrium position, the deformation of each of the springs is described by a function whose average value is different from zero.

In contrast to an oscillatory circuit with a single capacitor, where during oscillations its repetitive full recharge occurs, in the system considered, the initially charged capacitor is not completely recharged. For example, when its charge decreases to zero, and then is restored again in the same polarity. Otherwise, these oscillations do not differ from harmonic oscillations in a conventional circuit. The energy of these oscillations is conserved, if, of course, the resistance of the coil and connecting wires can be neglected.

Explain why, from a comparison of formulas (1) and (2) for mechanical and electromagnetic energies, it was concluded that the analog of the stiffness k is and the analog of the mass is the inductance and not vice versa.

Give justification for the derivation of expression (4) for the natural frequency of electromagnetic oscillations in the circuit from the analogy with a mechanical spring oscillator.

Harmonic oscillations in the -circuit are characterized by amplitude, frequency, period, oscillation phase, initial phase. Which of these quantities are determined by the properties of the oscillatory circuit itself, and which ones depend on the method of excitation of the oscillations?

Prove that the average values of electric and magnetic energies during natural oscillations in the circuit are equal to each other and make up half of the total electromagnetic energy of oscillations.

How to apply the laws of quasi-stationary phenomena in an electric circuit to derive a differential equation (12) for harmonic oscillations in a -circuit?

What differential equation does the current in an LC circuit satisfy?

Derive an equation for the rate of decrease in the energy of vibrations at low damping in the same way as it was done for a mechanical oscillator with friction proportional to the speed, and show that for time intervals significantly exceeding the oscillation period, this decrease occurs according to an exponential law. What is the meaning of the term "small attenuation" used here?

Show that the function given by formula (21) satisfies equation (16) for any values of and a.

Consider the mechanical system shown in Fig. 163, and find the dependence on the deformation time of the left spring and the speed of the massive body.

Loop without resistance with inevitable losses. In the problem considered above, despite the not quite usual initial conditions for charges on capacitors, it was possible to apply the usual equations for electrical circuits, since the conditions for the quasi-stationarity of the ongoing processes were satisfied there. But in the circuit, the diagram of which is shown in Fig. 164, with a formal external resemblance to the diagram in fig. 162, the conditions of quasi-stationarity are not satisfied if at the initial moment one capacitor is charged, and the second is not.

Let us discuss in more detail the reasons why the conditions of quasi-stationarity are violated here. Immediately after closing

Rice. 164. Electric circuit for which the conditions of quasi-stationarity are not met

The key is that all processes are played out only in interconnected capacitors, since the increase in current through the inductor is relatively slow and at first the branching of the current into the coil can be neglected.

When the key is closed, fast damped oscillations occur in a circuit consisting of capacitors and wires connecting them. The period of such oscillations is very small, since the inductance of the connecting wires is small. As a result of these oscillations, the charge on the capacitor plates is redistributed, after which the two capacitors can be considered as one. But at the first moment this cannot be done, because along with the redistribution of charges, there is also a redistribution of energy, part of which goes into heat.

After the damping of fast oscillations, oscillations occur in the system, as in a circuit with one capacitance capacitor, the charge of which at the initial moment is equal to the initial charge of the capacitor. The condition for the validity of the above reasoning is the smallness of the inductance of the connecting wires compared to the inductance of the coil.

As in the considered problem, it is useful to find a mechanical analogy here as well. If there the two springs corresponding to the condensers were located on either side of a massive body, here they must be located on one side of it, so that the vibrations of one of them can be transmitted to the other while the body is stationary. Instead of two springs, you can take one, but only at the initial moment it should be deformed inhomogeneously.

We grab the spring by the middle and stretch its left half for some distance. The second half of the spring will remain in an undeformed state, so that the load at the initial moment is displaced from the equilibrium position to the right by a distance and rests. Then let's release the spring. What features will result from the fact that at the initial moment the spring is deformed inhomogeneously? for, as it is easy to see, the stiffness of the “half” of the spring is If the mass of the spring is small compared to the mass of the ball, the natural frequency of the spring as an extended system is much greater than the frequency of the ball on the spring. These "fast" oscillations will die out in a time that is a small fraction of the period of the ball's oscillations. After the damping of fast oscillations, the tension in the spring is redistributed, and the displacement of the load remains practically the same, since the load does not have time to noticeably move during this time. The deformation of the spring becomes uniform, and the energy of the system is equal to

Thus, the role of fast oscillations of the spring was reduced to the fact that the energy reserve of the system decreased to the value that corresponds to the uniform initial deformation of the spring. It is clear that further processes in the system do not differ from the case of a homogeneous initial deformation. The dependence of the load displacement on time is expressed by the same formula (36).

In the example considered, as a result of rapid fluctuations, it turned into internal energy(into heat) half of the initial supply of mechanical energy. It is clear that by subjecting the initial deformation not to half, but to an arbitrary part of the spring, it is possible to convert any fraction of the initial supply of mechanical energy into internal energy. But in all cases, the energy of vibrations of the load on the spring corresponds to the energy reserve for the same uniform initial deformation of the spring.

In an electric circuit, as a result of damped fast oscillations, the energy of a charged capacitor is partially released in the form of Joule heat in the connecting wires. With equal capacities, this will be half the initial energy reserve. The other half remains in the form of energy of relatively slow electromagnetic oscillations in a circuit consisting of a coil and two capacitors C connected in parallel, and

Thus, in this system, idealization is fundamentally unacceptable, in which the dissipation of the oscillation energy is neglected. The reason for this is that rapid oscillations are possible here, without affecting the inductors or the massive body in a similar mechanical system.

Oscillatory circuit with non-linear elements. In studying mechanical vibrations, we have seen that vibrations are by no means always harmonic. Harmonic vibrations are characteristic property linear systems, in which

the restoring force is proportional to the deviation from the equilibrium position, and the potential energy is proportional to the square of the deviation. Real mechanical systems, as a rule, do not possess these properties, and oscillations in them can be considered harmonic only for small deviations from the equilibrium position.

In the case of electromagnetic oscillations in a circuit, one may get the impression that we are dealing with ideal systems in which the oscillations are strictly harmonic. However, this is true only as long as the capacitance of the capacitor and the inductance of the coil can be considered constant, i.e., independent of charge and current. A capacitor with a dielectric and a coil with a core are, strictly speaking, non-linear elements. When the capacitor is filled with a ferroelectric, i.e., a substance whose dielectric constant depends strongly on the applied electric field, the capacitance of the capacitor can no longer be considered constant. Similarly, the inductance of a coil with a ferromagnetic core depends on the strength of the current, since a ferromagnet has the property of magnetic saturation.

If in mechanical oscillatory systems the mass, as a rule, can be considered constant and nonlinearity occurs only due to the nonlinear nature of the acting force, then in an electromagnetic oscillatory circuit, nonlinearity can occur both due to a capacitor (analogous to an elastic spring) and due to an inductor ( mass analogue).

Why is idealization inapplicable for an oscillatory circuit with two parallel capacitors (Fig. 164), in which the system is considered conservative?

Why are the fast oscillations leading to the dissipation of the oscillation energy in the circuit in Fig. 164 did not occur in the circuit with two series capacitors shown in fig. 162?

What reasons can lead to non-sinusoidality of electromagnetic oscillations in the circuit?

An electrical oscillatory circuit is a system for excitation and maintenance of electromagnetic oscillations. In its simplest form, this is a circuit consisting of a coil with an inductance L, a capacitor with a capacitance C and a resistor with a resistance R connected in series (Fig. 129). When switch P is set to position 1, capacitor C is charged to a voltage U t. In this case, between the plates of the capacitor is formed electric field, whose maximum energy is equal to

When the switch is moved to position 2, the circuit closes and the following processes take place in it. The capacitor begins to discharge and current flows through the circuit i, the value of which increases from zero to the maximum value and then decreases back to zero. Since an alternating current flows in the circuit, an EMF is induced in the coil, which prevents the capacitor from discharging. Therefore, the process of discharging the capacitor does not occur instantly, but gradually. As a result of the appearance of current in the coil, a magnetic field arises, the energy of which is
reaches its maximum value at a current equal to . The maximum energy of the magnetic field will be equal to

After reaching the maximum value, the current in the circuit will begin to decrease. In this case, the capacitor will be recharged, the energy of the magnetic field in the coil will decrease, and the energy of the electric field in the capacitor will increase. Upon reaching the maximum value. The process will begin to repeat and oscillations of electric and magnetic fields occur in the circuit. If we assume that the resistance
(i.e. no energy is spent on heating), then according to the law of conservation of energy, the total energy W remains constant

and
;
.

A circuit in which there is no energy loss is called ideal. The voltage and current in the circuit change according to the harmonic law

;

where - circular (cyclic) oscillation frequency
.

The circular frequency is related to the oscillation frequency and periods of fluctuations T ratio.

H and fig. 130 shows graphs of voltage U and current I in the coil of an ideal oscillatory circuit. It can be seen that the current strength lags in phase with the voltage by .

;
;
- Thomson's formula.

In the event that the resistance
, the Thomson formula takes the form

Fundamentals of Maxwell's theory

Maxwell's theory is the theory of a single electromagnetic field created by an arbitrary system of charges and currents. In theory, the main problem of electrodynamics is solved - according to a given distribution of charges and currents, the characteristics of the electric and magnetic fields created by them are found. Maxwell's theory is a generalization of the most important laws describing electrical and electromagnetic phenomena - the Ostrogradsky-Gauss theorem for electric and magnetic fields, the law of total current, the law electromagnetic induction and theorems on the circulation of the electric field strength vector. Maxwell's theory is phenomenological in nature, i.e. it does not consider the internal mechanism of phenomena occurring in the environment and causing the appearance electric and magnetic fields. In Maxwell's theory, the medium is described using three characteristics - dielectric ε and magnetic μ permeability of the medium and electrical conductivity γ.

Electrical oscillations are understood as periodic changes in charge, current and voltage. The simplest system in which free electrical oscillations are possible is the so-called oscillatory circuit. This is a device consisting of a capacitor and a coil connected to each other. We will assume that there is no active resistance of the coil, in this case the circuit is called ideal. When energy is communicated to this system, undamped harmonic oscillations of the charge on the capacitor, voltage and current will occur in it.

It is possible to inform the oscillatory circuit of energy different ways. For example, by charging a capacitor from a source direct current or excitation current in the inductor. In the first case, the electric field between the plates of the capacitor possesses energy. In the second, the energy is contained in the magnetic field of the current flowing through the circuit.

§1 The equation of oscillations in the circuit

Let us prove that when energy is imparted to the circuit, undamped harmonic oscillations will occur in it. To do this, it is necessary to obtain a differential equation of harmonic oscillations of the form .

Suppose the capacitor is charged and closed to the coil. The capacitor will begin to discharge, current will flow through the coil. According to Kirchhoff's second law, the sum of voltage drops along a closed circuit is equal to the sum of the EMF in this circuit .

In our case, the voltage drop is because the circuit is ideal. The capacitor in the circuit behaves like a current source, the potential difference between the capacitor plates acts as an EMF, where is the charge on the capacitor, is the capacitance of the capacitor. In addition, when a changing current flows through the coil, an EMF of self-induction arises in it, where is the inductance of the coil, is the rate of change of current in the coil. Since the EMF of self-induction prevents the process of discharging the capacitor, the second Kirchhoff law takes the form

But the current in the circuit is the current of discharging or charging the capacitor, therefore. Then

The differential equation is transformed to the form

By introducing the notation , we obtain the well-known differential equation of harmonic oscillations.

This means that the charge on the capacitor in the oscillatory circuit will change according to the harmonic law

where is the maximum value of the charge on the capacitor, is the cyclic frequency, is the initial phase of the oscillations.

Charge oscillation period . This expression is called the Thompson formula.

Capacitor voltage

Circuit current

We see that in addition to the charge on the capacitor, according to the harmonic law, the current in the circuit and the voltage on the capacitor will also change. The voltage oscillates in phase with the charge, and the current is ahead of the charge in

phase on .

Capacitor electric field energy

The energy of the magnetic field current

Thus, the energies of the electric and magnetic fields also change according to the harmonic law, but with a doubled frequency.

Summarize

Electric oscillations should be understood as periodic changes in charge, voltage, current strength, electric field energy, magnetic field energy. These oscillations, like mechanical ones, can be both free and forced, harmonic and non-harmonic. Free harmonic electrical oscillations are possible in an ideal oscillatory circuit.

§2 Processes occurring in an oscillatory circuit

We mathematically proved the existence of free harmonic oscillations in an oscillatory circuit. However, it remains unclear why such a process is possible. What causes oscillations in a circuit?

In the case of free mechanical vibrations, such a reason was found - this is inner strength, which arises when the system is taken out of equilibrium. This force at any moment is directed to the equilibrium position and is proportional to the coordinate of the body (with a minus sign). Let's try to find a similar reason for the occurrence of oscillations in the oscillatory circuit.

Let the oscillations in the circuit excite by charging the capacitor and closing it to the coil.

At the initial moment of time, the charge on the capacitor is maximum. Consequently, the voltage and energy of the electric field of the capacitor are also maximum.

There is no current in the circuit, the energy of the magnetic field of the current is zero.

First quarter of the period- capacitor discharge.

The capacitor plates, having different potentials, are connected by a conductor, so the capacitor begins to discharge through the coil. The charge, the voltage on the capacitor and the energy of the electric field decrease.

The current that appears in the circuit increases, however, its growth is prevented by the self-induction EMF that occurs in the coil. The energy of the magnetic field of the current increases.

A quarter has passed- the capacitor is discharged.

The capacitor discharged, the voltage across it became equal to zero. The energy of the electric field at this moment is also equal to zero. According to the law of conservation of energy, it could not disappear. The energy of the field of the capacitor has completely turned into the energy of the magnetic field of the coil, which at this moment reaches its maximum value. The maximum current in the circuit.

It would seem that at this moment the current in the circuit should stop, because the cause of the current, the electric field, has disappeared. However, the disappearance of the current is again prevented by the EMF of self-induction in the coil. Now it will maintain a decreasing current, and it will continue to flow in the same direction, charging the capacitor. The second quarter of the period begins.

Second quarter of the period - Capacitor recharge.

The current supported by the self-induction EMF continues to flow in the same direction, gradually decreasing. This current charges the capacitor in opposite polarity. The charge and voltage across the capacitor increase.

The energy of the magnetic field of the current, decreasing, passes into the energy of the electric field of the capacitor.

The second quarter of the period has passed - the capacitor has recharged.

The capacitor recharges as long as there is current. Therefore, at the moment when the current stops, the charge and voltage on the capacitor take on a maximum value.

The energy of the magnetic field at this moment completely turned into the energy of the electric field of the capacitor.

The situation in the circuit at this moment is equivalent to the original one. The processes in the circuit will be repeated, but in the opposite direction. One complete oscillation in the circuit, lasting for a period, will end when the system returns to its original state, that is, when the capacitor is recharged in its original polarity.

It is easy to see that the cause of oscillations in the circuit is the phenomenon of self-induction. The EMF of self-induction prevents a change in current: it does not allow it to instantly increase and instantly disappear.

By the way, it would not be superfluous to compare the expressions for calculating the quasi-elastic force in a mechanical oscillatory system and the EMF of self-induction in the circuit:

Previously, differential equations were obtained for mechanical and electrical oscillatory systems:

In spite of fundamental differences physical processes to mechanical and electrical oscillatory systems, the mathematical identity of the equations describing the processes in these systems is clearly visible. This should be discussed in more detail.

§3 Analogy between electrical and mechanical vibrations

A careful analysis of the differential equations for a spring pendulum and an oscillatory circuit, as well as formulas relating the quantities characterizing the processes in these systems, makes it possible to identify which quantities behave in the same way (Table 2).

Spring pendulum	Oscillatory circuit
Body coordinate ()	Charge on the capacitor ()
body speed	Loop current
Potential energy of an elastically deformed spring	Capacitor electric field energy
Kinetic energy of the load	The energy of the magnetic field of the coil with current
The reciprocal of the spring stiffness	Capacitor capacity
Load weight	Coil inductance
Elastic force	EMF of self-induction, equal to the voltage on the capacitor

table 2

It is important not just a formal similarity between the quantities that describe the processes of pendulum oscillation and the processes in the circuit. The processes themselves are identical!

The extreme positions of the pendulum are equivalent to the state of the circuit when the charge on the capacitor is maximum.

The equilibrium position of the pendulum is equivalent to the state of the circuit when the capacitor is discharged. At this moment, the elastic force vanishes, and there is no voltage on the capacitor in the circuit. The speed of the pendulum and the current in the circuit are maximum. The potential energy of elastic deformation of the spring and the energy of the electric field of the capacitor are equal to zero. The energy of the system consists of the kinetic energy of the load or the energy of the magnetic field of the current.

The discharge of the capacitor proceeds similarly to the movement of the pendulum from extreme position to a position of balance. The process of recharging the capacitor is identical to the process of removing the load from the equilibrium position to the extreme position.

Total energy of the oscillatory system or remains unchanged over time.

A similar analogy can be traced not only between a spring pendulum and an oscillatory circuit. General patterns of free oscillations of any nature! These patterns, illustrated by the example of two oscillatory systems (a spring pendulum and an oscillatory circuit), are not only possible, but must see in the vibrations of any system.

In principle, it is possible to solve the problem of any oscillatory process by replacing it with pendulum oscillations. To do this, it is enough to competently build an equivalent mechanical system, solve a mechanical problem and change the values in the final result. For example, you need to find the period of oscillation in a circuit containing a capacitor and two coils connected in parallel.

The oscillatory circuit contains one capacitor and two coils. Since the coil behaves like the weight of a spring pendulum and the capacitor behaves like a spring, the equivalent mechanical system must contain one spring and two weights. The whole problem is how the weights are attached to the spring. Two cases are possible: one end of the spring is fixed, and one weight is attached to the free end, the second is on the first one, or the weights are attached to different ends of the spring.

At parallel connection coils of different inductance currents flow through them different. Consequently, the speeds of the loads in an identical mechanical system must also be different. Obviously, this is possible only in the second case.

We have already found the period of this oscillatory system. He is equal . Replacing the masses of the weights by the inductance of the coils, and the reciprocal of the spring stiffness by the capacitance of the capacitor, we obtain .

§4 Oscillatory circuit with a direct current source

Consider an oscillatory circuit containing a direct current source. Let the capacitor be initially uncharged. What will happen in the system after the key K is closed? Will oscillations be observed in this case and what is their frequency and amplitude?

Obviously, after the key is closed, the capacitor will begin to charge. We write Kirchhoff's second law:

The current in the circuit is the charging current of the capacitor, therefore. Then . The differential equation is transformed to the form

*Solve the equation by change of variables.

Let's denote . Differentiate twice and, taking into account that , we obtain . The differential equation takes the form

This is a differential equation of harmonic oscillations, its solution is the function

where is the cyclic frequency, the integration constants and are found from the initial conditions.

The charge on a capacitor changes according to the law

Immediately after the switch is closed, the charge on the capacitor zero and there is no current in the circuit . Taking into account the initial conditions, we obtain a system of equations:

Solving the system, we get and . After the key is closed, the charge on the capacitor changes according to the law.

It is easy to see that harmonic oscillations occur in the circuit. The presence of a direct current source in the circuit did not affect the oscillation frequency, it remained equal. The “equilibrium position” has changed - at the moment when the current in the circuit is maximum, the capacitor is charged. The amplitude of the charge oscillations on the capacitor is equal to Cε.

The same result can be obtained more simply by using the analogy between oscillations in a circuit and oscillations of a spring pendulum. DC source is equivalent to DC force field, in which a spring pendulum is placed, for example, a gravitational field. The absence of charge on the capacitor at the moment of closing the circuit is identical to the absence of deformation of the spring at the moment of bringing the pendulum into oscillatory motion.

In a constant force field, the period of oscillation of a spring pendulum does not change. The oscillation period in the circuit behaves in the same way - it remains unchanged when a direct current source is introduced into the circuit.

In the equilibrium position, when the load speed is maximum, the spring is deformed:

When the current in the oscillatory circuit is maximum . Kirchhoff's second law is written as follows

At this moment, the charge on the capacitor is equal to The same result could be obtained based on the expression (*) by replacing

§5 Examples of problem solving

Task 1 Law of energy conservation

L\u003d 0.5 μH and a capacitor with a capacitance With= 20 pF electrical oscillations occur. What is the maximum voltage across the capacitor if the amplitude of the current in the circuit is 1 mA? The active resistance of the coil is negligible.

Decision:

(1)

2 At the moment when the voltage on the capacitor is maximum (maximum charge on the capacitor), there is no current in the circuit. The total energy of the system consists only of the energy of the electric field of the capacitor

(2)

3 At the moment when the current in the circuit is maximum, the capacitor is completely discharged. The total energy of the system consists only of the energy of the magnetic field of the coil

(3)

4 Based on expressions (1), (2), (3), we obtain the equality . The maximum voltage across the capacitor is

Task 2 Law of energy conservation

In an oscillatory circuit consisting of an inductance coil L and a capacitor WITH, electrical oscillations occur with a period T = 1 μs. Maximum charge value . What is the current in the circuit at the moment when the charge on the capacitor is equal to? The active resistance of the coil is negligible.

Decision:

1 Since the active resistance of the coil can be neglected, the total energy of the system, consisting of the energy of the electric field of the capacitor and the energy of the magnetic field of the coil, remains unchanged over time:

(1)

2 At the moment when the charge on the capacitor is maximum, there is no current in the circuit. The total energy of the system consists only of the energy of the electric field of the capacitor

(2)

3 Based on (1) and (2), we obtain the equality . The current in the circuit is .

4 The oscillation period in the circuit is determined by the Thomson formula. From here. Then for the current in the circuit we obtain

Task 3 Oscillatory circuit with two capacitors connected in parallel

In an oscillatory circuit consisting of an inductance coil L and a capacitor WITH, electrical oscillations occur with an amplitude of charge. At the moment when the charge on the capacitor is maximum, the key K is closed. What will be the period of oscillations in the circuit after the key is closed? What is the amplitude of the current in the circuit after closing the switch? Ignore the ohmic resistance of the circuit.

Decision:

1 Closing the key leads to the appearance in the circuit of another capacitor connected in parallel to the first one. The total capacitance of two capacitors connected in parallel is .

The period of oscillations in the circuit depends only on its parameters and does not depend on how oscillations were excited in the system and what energy was imparted to the system for this. According to the Thomson formula.

2 To find the amplitude of the current, let's find out what processes occur in the circuit after the key is closed.

The second capacitor was connected at the moment when the charge on the first capacitor was maximum, therefore, there was no current in the circuit.

The loop capacitor should begin to discharge. The discharge current, having reached the node, should be divided into two parts. However, in the branch with the coil, an EMF of self-induction occurs, which prevents the increase in the discharge current. For this reason, the entire discharge current will flow into the branch with the capacitor, the ohmic resistance of which is zero. The current will stop as soon as the voltages on the capacitors are equal, while the initial charge of the capacitor is redistributed between the two capacitors. The charge redistribution time between two capacitors is negligible due to the absence of ohmic resistance in the capacitor branches. During this time, the current in the branch with the coil will not have time to appear. fluctuations in new system continue after the charge is redistributed between the capacitors.

It is important to understand that in the process of redistributing the charge between two capacitors, the energy of the system is not conserved! Before the key was closed, one capacitor, a loop capacitor, had energy:

After the charge is redistributed, a battery of capacitors possesses energy:

It is easy to see that the energy of the system has decreased!

3 We find the new amplitude of the current using the law of conservation of energy. In the process of oscillations, the energy of the capacitor bank is converted into the energy of the magnetic field of the current:

Please note that the law of conservation of energy begins to "work" only after the completion of the redistribution of charge between the capacitors.

Task 4 Oscillatory circuit with two capacitors connected in series

The oscillatory circuit consists of a coil with an inductance L and two capacitors C and 4C connected in series. A capacitor with a capacity of C is charged to a voltage, a capacitor with a capacity of 4C is not charged. After the key is closed, oscillations begin in the circuit. What is the period of these oscillations? Determine the amplitude of the current, the maximum and minimum voltage values on each capacitor.

Decision:

1 At the moment when the current in the circuit is maximum, there is no self-induction EMF in the coil . We write down for this moment the second law of Kirchhoff

We see that at the moment when the current in the circuit is maximum, the capacitors are charged to the same voltage, but in the opposite polarity:

2 Before closing the key, the total energy of the system consisted only of the energy of the electric field of the capacitor C:

At the moment when the current in the circuit is maximum, the energy of the system is the sum of the energy of the magnetic field of the current and the energy of two capacitors charged to the same voltage:

According to the law of conservation of energy

To find the voltage on the capacitors, we use the law of conservation of charge - the charge of the lower plate of the capacitor C has partially transferred to the upper plate of the capacitor 4C:

We substitute the found voltage value into the law of conservation of energy and find the amplitude of the current in the circuit:

3 Let's find the limits within which the voltage on the capacitors changes during the oscillation process.

It is clear that at the moment the circuit was closed, there was a maximum voltage on the capacitor C. Capacitor 4C was not charged, therefore, .

After the switch is closed, capacitor C begins to discharge, and a capacitor with a capacity of 4C begins to charge. The process of discharging the first and charging the second capacitors ends as soon as the current in the circuit stops. This will happen in half a period. According to the laws of conservation of energy and electric charge:

Solving the system, we find:

The minus sign means that after half a period, the capacitance C is charged in the reverse polarity of the original.

Task 5 Oscillatory circuit with two coils connected in series

The oscillating circuit consists of a capacitor with a capacitance C and two coils with an inductance L1 and L2. At the moment when the current in the circuit has reached its maximum value, an iron core is quickly introduced into the first coil (compared to the oscillation period), which leads to an increase in its inductance by μ times. What is the voltage amplitude in the process of further oscillations in the circuit?

Decision:

1 With the rapid introduction of the core into the coil, the magnetic flux(the phenomenon of electromagnetic induction). Therefore, a rapid change in the inductance of one of the coils will result in a rapid change in the current in the circuit.

2 During the introduction of the core into the coil, the charge on the capacitor did not have time to change, it remained uncharged (the core was introduced at the moment when the current in the circuit was maximum). After a quarter of the period, the energy of the magnetic field of the current will turn into the energy of a charged capacitor:

Substitute in the resulting expression the value of the current I and find the amplitude of the voltage across the capacitor:

Task 6 Oscillatory circuit with two coils connected in parallel

The inductors L 1 and L 2 are connected through the keys K1 and K2 to a capacitor with a capacitance C. At the initial moment, both keys are open, and the capacitor is charged to a potential difference. First, the key K1 is closed and, when the voltage across the capacitor becomes equal to zero, K2 is closed. Determine the maximum voltage across the capacitor after closing K2. Ignore coil resistances.

Decision:

1 When the key K2 is open, oscillations occur in the circuit consisting of the capacitor and the first coil. By the time K2 is closed, the energy of the capacitor has transferred into the energy of the magnetic field of the current in the first coil:

2 After closing K2, two coils connected in parallel appear in the oscillatory circuit.

The current in the first coil cannot stop due to the phenomenon of self-induction. At the node, it divides: one part of the current goes to the second coil, and the other part charges the capacitor.

3 The voltage on the capacitor will become maximum when the current stops I charging capacitor. It is obvious that at this moment the currents in the coils will be equal.

: The weights are subject to the same modulus of force - both weights are attached to the spring Immediately after the closure of K2, a current existed in the first coil At the initial moment, the first load had a speed Immediately after closing K2, there was no current in the second coil At the initial moment, the second load was at rest What is the maximum voltage across the capacitor? What is the maximum elastic force that occurs in the spring during oscillation?

The pendulum moves forward with the speed of the center of mass and oscillates about the center of mass.

The elastic force is maximum at the moment of maximum deformation of the spring. Obviously, at this moment, the relative speed of the weights becomes equal to zero, and relative to the table, the weights move at the speed of the center of mass. We write down the law of conservation of energy:

Solving the system, we find

We make a replacement

and get for maximum voltage previously found value

§6 Tasks for independent solution

Exercise 1 Calculation of the period and frequency of natural oscillations

1 The oscillatory circuit includes a coil of variable inductance, varying within L1= 0.5 µH to L2\u003d 10 μH, and a capacitor, the capacitance of which can vary from From 1= 10 pF to

From 2\u003d 500 pF. What frequency range can be covered by tuning this circuit?

2 How many times will the frequency of natural oscillations in the circuit change if its inductance is increased by 10 times, and the capacitance is reduced by 2.5 times?

3 An oscillatory circuit with a 1 uF capacitor is tuned to a frequency of 400 Hz. If you connect a second capacitor in parallel to it, then the oscillation frequency in the circuit becomes equal to 200 Hz. Determine the capacitance of the second capacitor.

4 The oscillatory circuit consists of a coil and a capacitor. How many times will the frequency of natural oscillations in the circuit change if a second capacitor is connected in series in the circuit, the capacitance of which is 3 times less than the capacitance of the first?

5 Determine the oscillation period of the circuit, which includes a coil (without core) of length in= 50 cm m cross-sectional area

S\u003d 3 cm 2, having N\u003d 1000 turns, and a capacitance capacitor With= 0.5 uF.

6 The oscillatory circuit includes an inductor L\u003d 1.0 μH and an air capacitor, the areas of the plates of which S\u003d 100 cm 2. The circuit is tuned to a frequency of 30 MHz. Determine the distance between the plates. The active resistance of the circuit is negligible.

ELECTROMAGNETIC OSCILLATIONS AND WAVES

§1 Oscillatory circuit.

Natural vibrations in the oscillatory circuit.

Thomson formula.

Damped and forced oscillations in the c.c.

Free vibrations in c.c.

An oscillatory circuit (c.c.) is a circuit consisting of a capacitor and an inductor. Under certain conditions in the c.c. electromagnetic fluctuations in charge, current, voltage and energy can occur.

Consider the circuit shown in Figure 2. If you put the key in position 1, then the capacitor will be charged and a charge will appear on its platesQ and tension U C. If you then turn the key to position 2, then the capacitor will begin to discharge, current will flow in the circuit, while the energy of the electric field enclosed between the plates of the capacitor will be converted into magnetic field energy concentrated in the inductorL. The presence of an inductor leads to the fact that the current in the circuit does not increase instantly, but gradually due to the phenomenon of self-induction. As the capacitor discharges, the charge on its plates will decrease, the current in the circuit will increase. The maximum value of the loop current will reach when the charge on the plates is equal to zero. From this point on, the loop current will begin to decrease, but, due to the phenomenon of self-induction, it will be maintained by the magnetic field of the inductor, i.e. when the capacitor is fully discharged, the energy of the magnetic field stored in the inductor will begin to turn into the energy of an electric field. Because of the loop current, the capacitor will begin to recharge and a charge opposite to the original one will begin to accumulate on its plates. The capacitor will be recharged until all the energy of the magnetic field of the inductor is converted into the energy of the electric field of the capacitor. Then the process will be repeated in the opposite direction, and thus, electromagnetic oscillations will occur in the circuit.

Let us write down the 2nd Kirchhoff's law for the considered k.k.,

Differential equation k.k.

We have obtained a differential equation for charge oscillations in a c.c. This equation is similar to a differential equation describing the motion of a body under the action of a quasi-elastic force. Therefore, the solution of this equation will be written similarly

The equation of charge fluctuations in c.c.

The equation of voltage fluctuations on the capacitor plates in the c.c.

The equation of current fluctuations in k.k.

Damped oscillations in QC

Consider a C.C. containing capacitance, inductance, and resistance. Kirchhoff's 2nd law in this case will be written in the form

- attenuation factor,

Own cyclic frequency.

- - differential equation of damped oscillations in the c.c.

The equation of damped charge oscillations in a c.c.

The law of change of the charge amplitude during damped oscillations in the c.c.;

The period of damped oscillations.

Decrement of attenuation.

- logarithmic damping decrement.

The goodness of the circuit.

If damping is weak, then T ≈T 0

We investigate the change in voltage on the capacitor plates.

The change in current is out of phase by φ from the voltage.

at - damped oscillations are possible,

at - critical situation

at , i.e. R > RTo- fluctuations do not occur (aperiodic discharge of the capacitor).

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

Free are called such fluctuations, which arise in a closed system due to the deviation of this system from a state of stable equilibrium.

During oscillations, a continuous process of transformation of the energy of the system from one form into another takes place. In case of hesitation electromagnetic field the 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 oscillatory circuit (LC circuit) - an electrical circuit consisting of an inductance coil L and a capacitor C.

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

Figure 1 shows a diagram of an ideal oscillatory circuit.

Circuit energy

Total energy of the oscillatory 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 We- energy of the electric field of the oscillatory circuit in this moment time With is the capacitance of the capacitor, u- the value of the voltage on the capacitor at a given time, q- the value of the charge of the capacitor at a given time, Wm- the energy of the magnetic field of the oscillatory circuit at a given time, L- coil inductance, i- the value of the current in the coil at a given time.

Processes in the oscillatory circuit

Consider the processes that occur in the oscillatory circuit.

To remove the circuit from the equilibrium position, we charge the capacitor so that there is 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 point in time, i.e. i = 0.

After the key is closed, under the action of the electric field of the capacitor in the circuit, electricity, current strength i which will increase over time. The capacitor at this time will begin to discharge, because. the electrons that create the current (I remind you that the direction of the 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 ). Along with charge q tension will decrease u\(\left(u = \dfrac(q)(C) \right).\) As the current strength increases, a self-induction emf will appear through the coil, preventing a change in the current strength. As a result, the current strength in the oscillatory circuit will increase from zero to a certain maximum value not instantly, but over 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 electric field of the capacitor (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 give it a negative charge, the electrons leaving the neutral plate give it a positive charge. The capacitor begins to charge q(and voltage u), but of opposite sign, i.e. the capacitor is recharged. Now the new electric field of the capacitor prevents the electrons from moving, so the current i begins to decrease (see Fig. 2, position 4 ). Again, this does not happen instantly, since now the self-induction EMF seeks to compensate for the decrease in current and “supports” it. And the value of the current I m(pregnant 3 ) turns out maximum current in 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 We\(\left(W_(e)=\dfrac(q^(2))(2C)=\dfrac(C \cdot u^(2))(2) \right),\) 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, the 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 oscillatory circuit is conserved over time, since there is energy loss in it (no resistance). Then

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

Thus, ideally LC- the circuit will experience periodic changes in current strength values i, charge q and stress u, and the total energy of the circuit will remain constant. In this case, we say that there are 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 consuming energy from external sources.

Thus, the occurrence of free electromagnetic oscillations in the circuit is due to the recharging of the capacitor and the occurrence of self-induction EMF in the coil, which “provides” this recharging. Note that the charge on the capacitor q and the current in the coil i reach their maximum values Q m and I m at various points in time.

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 lining), is called the period of free (natural) electromagnetic oscillations in the circuit.

The period of free electromagnetic oscillations in LC-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 mechanical analogy, an ideal oscillatory circuit corresponds to a spring pendulum without friction, and a real one - with friction. Due to the action of friction forces, the oscillations of a spring pendulum damp out over time.

*Derivation of the Thomson formula

Since the total energy of the ideal LC-contour, equal to the sum of energies electrostatic field capacitor and the magnetic field of the coil is preserved, then at any 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-circuit, using the law of conservation of energy. Differentiating the expression for its total energy with respect to time, taking into account the fact that

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

we obtain an equation describing free oscillations in an ideal circuit:

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

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

By rewriting it as:

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

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 oscillations under consideration

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

Literature

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