Equation of current strength in an oscillatory circuit. Equation describing processes in an oscillatory circuit

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

• Free these are called fluctuations, which arise in a closed system as a result of deviation of this system from a state of stable equilibrium.

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

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

Unlike a real oscillatory circuit, which has electrical resistance R, the electrical resistance of an ideal circuit 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 energies

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 W e- energy of the electric field of the oscillatory circuit at a given time, WITH- electrical capacity of the capacitor, u- the voltage value on the capacitor at a given time, q- value of the capacitor charge at a given time, W m- 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 an oscillatory circuit

Let us consider the processes that occur in an oscillatory circuit.

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

After closing the key under the action of the electric field of the capacitor, a electricity, current strength i which will increase over time. The capacitor will begin to discharge at this time, because electrons creating a current (I remind you that the direction of current is taken to be the direction of movement of positive charges) leave the negative plate of the capacitor and come to the positive one (see Fig. 2, position 2 ). Along with charge q tension will also decrease u\(\left(u = \dfrac(q)(C) \right).\) When the current strength increases through the coil, a self-induction emf will arise, which prevents the current from changing. As a result, the current strength in the oscillating 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 creating the current continue to move by inertia. In this case, electrons arriving at the neutral plate of the capacitor impart a negative charge to it, and 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 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 tends to compensate for the decrease in current and “supports” it. And the current value I m(pregnant 3 ) turns out maximum current value in the circuit.

And again, under the influence 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 ). And so on.

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 strength 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, energy 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 because there is no energy loss (no resistance). Then

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

Thus, in an ideal L.C.- the circuit will undergo periodic changes in current values i, charge q and voltage u, and the total energy of the circuit will remain constant. In this case, they say that there are problems in the circuit free electromagnetic oscillations.

• Free electromagnetic oscillations in the circuit - these are periodic changes in the charge on the capacitor plates, current 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 a self-inductive emf in the coil, which “provides” this recharging. Note that the capacitor charge q and the current in the coil i reach their maximum values Qm 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 shortest period of time during which L.C.- the circuit returns to its original state (to the initial value of the charge of a given plate), called the period of free (natural) electromagnetic oscillations in the circuit.

The period of free electromagnetic oscillations in L.C.-contour is determined by Thomson’s formula:

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

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

*Derivation of Thomson's formula

Since the total energy of the ideal L.C.-circuit equal to the sum of the energies of the electrostatic field of the capacitor and the magnetic field of the coil is conserved, then at any time the equality is valid

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

We obtain the equation of oscillations in L.C.-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.\ )

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 oscillations

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

Literature

1. Zhilko, V.V. Physics: textbook. manual for 11th grade general education. school from Russian language training / V.V. Zhilko, L.G. Markovich. - Minsk: Nar. Asveta, 2009. - pp. 39-43.

A circuit that consists of a coil of inductance L and a capacitor of capacitance C connected in series is called an oscillatory circuit.

2. Why is the total energy of the electromagnetic field conserved in an oscillatory circuit?

Because it is not spent on heating (R ≈ 0).

3. Explain why harmonic, undamped oscillations of charge and current occur in the circuit.

At the initial moment t = 0, an electric field is formed between the plates of the capacitor. At time t = T/4, the current in the circuit decreases, and the magnetic flux in the coil decreases. The capacitor begins to recharge, and an electric field appears between its plates, which tends to reduce the current. At time t = T/2, the current is 0. The charge on the plates is equal to the original in absolute value, but opposite in direction. Then all processes will begin to flow in the opposite direction, and at the moment t = T the system will return to its original state. The cycle will then repeat. In the circuit, in the absence of losses due to heating of the wires, harmonic undamped oscillations of the charge on the capacitor plates and the current strength in the inductors occur.

4. According to what law do the charge on the capacitor and the current in the inductor change over time?

According to Ohm's law for an oscillatory circuit.

5. How does the period of natural oscillations in the oscillatory circuit depend on the value of the electrical capacitance of the capacitor and the inductance of the coil?

ELECTROMAGNETIC OSCILLATIONS AND WAVES

§1 Oscillatory circuit.

Natural vibrations in an oscillatory circuit.

Thomson's formula.

Damped and forced oscillations in k.k.

1. Free oscillations in k.k.

An oscillating circuit (OC) is a circuit consisting of a capacitor and an inductor. Under certain conditions in the k.k. Electromagnetic fluctuations of charge, current, voltage and energy may occur.

Consider the circuit shown in Fig. 2. If you put the key in position 1, the capacitor will charge and a charge will appear on its platesQ and voltage U C. If you then move the key to position 2, the capacitor will begin to discharge, current will flow in the circuit, and the energy of the electric field contained 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, and the current in the circuit will increase. The circuit current will reach its maximum value when the charge on the plates is equal to zero. From this moment, the loop current will begin to decrease, but, due to the phenomenon of self-induction, it will be supported by the magnetic field of the inductor, i.e. When the capacitor is completely discharged, the energy of the magnetic field stored in the inductor will begin to transform into the energy of the electric field. Due to 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 arise in the circuit.

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

Differential equation k.k.

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

Equation of charge oscillations in k.k.

Equation of voltage oscillations on the capacitor plates in the s.c.c.

Equation of current oscillations in a c.c.

1. Damped oscillations in k.k.

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

- attenuation coefficient,

Natural cyclic frequency.

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

Equation of damped oscillations of a charge in a c.c.

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

Period of damped oscillations.

Decrement of attenuation.

- logarithmic damping decrement.

Contour quality factor.

If the attenuation is weak, then T ≈T 0

Let's study the change in voltage on the capacitor plates.

The change in current differs in phase by φ from the voltage.

at - damped oscillations are possible,

at - critical position

at , i.e. R > RTO- oscillations do not occur (aperiodic capacitor discharge).

Advances in the study of electromagnetism in the 19th century led to the rapid development of industry and technology, especially in communications. While laying telegraph lines over long distances, engineers encountered a number of inexplicable phenomena that prompted scientists to conduct research. So, in the 50s, the British physicist William Thomson (Lord Kelvin) took up the issue of transatlantic telegraphy. Taking into account the failures of the first practitioners, he theoretically investigated the issue of the propagation of electrical impulses along a cable. At the same time, Kelvin received a number of important conclusions, which later made it possible to implement telegraphy across the ocean. Also in 1853, a British physicist derived the conditions for the existence of an oscillatory electric discharge. These conditions formed the basis of the entire study of electrical oscillations. In this lesson and other lessons in this chapter, we will look at some basics of Thomson's theory of electrical oscillations.

Periodic or almost periodic changes in charge, current and voltage in a circuit are called electromagnetic vibrations. One more definition can also be given.

Electromagnetic vibrations are called periodic changes in the electric field strength ( E) and magnetic induction ( B).

To excite electromagnetic oscillations, it is necessary to have an oscillatory system. The simplest oscillatory system in which free electromagnetic oscillations can be maintained is called oscillatory circuit.

Figure 1 shows the simplest oscillatory circuit - this is an electrical circuit that consists of a capacitor and a conducting coil connected to the capacitor plates.

Rice. 1. Oscillatory circuit

Free electromagnetic oscillations can occur in such an oscillatory circuit.

Free are called oscillations that are carried out due to the energy reserves accumulated by the oscillatory system itself, without attracting energy from the outside.

Consider the oscillatory circuit shown in Figure 2. It consists of: a coil with inductance L, capacitor with capacitance C, a light bulb (to control the presence of current in the circuit), a key and a current source. Using a key, the capacitor can be connected either to a current source or to a coil. At the initial moment of time (the capacitor is not connected to a current source), the voltage between its plates is 0.

Rice. 2. Oscillatory circuit

We charge the capacitor by connecting it to a DC source.

When switching the capacitor to the coil, the light bulb turns on a short time lights up, that is, the capacitor is quickly discharged.

Rice. 3. Graph of the voltage between the capacitor plates versus time during discharge

Figure 3 shows a graph of the voltage between the capacitor plates versus time. This graph shows the time interval from the moment the capacitor is switched to the coil until the voltage across the capacitor is zero. It can be seen that the voltage changed periodically, that is, oscillations occurred in the circuit.

Consequently, free damped electromagnetic oscillations flow in the oscillatory circuit.

At the initial moment of time (before the capacitor was closed to the coil), all the energy was concentrated in the electric field of the capacitor (see Fig. 4 a).

When a capacitor is shorted to a coil, it will begin to discharge. The discharge current of the capacitor, passing through the turns of the coil, creates a magnetic field. This means that there is a change in the magnetic flux surrounding the coil, and a self-induction emf appears in it, which prevents the instantaneous discharge of the capacitor, therefore, the discharge current increases gradually. As the discharge current increases, the electric field in the capacitor decreases, but the magnetic field of the coil increases (see Fig. 4 b).

At the moment when the capacitor field disappears (the capacitor is discharged), the magnetic field of the coil will be maximum (see Fig. 4 c).

Further, the magnetic field will weaken and a self-induction current will appear in the circuit, which will prevent the magnetic field from decreasing; therefore, this self-induction current will be directed in the same way as the discharge current of the capacitor. This will cause the capacitor to recharge. That is, on the cover where there was a plus sign at first, a minus will appear, and vice versa. The direction of the electric field strength vector in the capacitor will also change to the opposite (see Fig. 4 d).

The current in the circuit will weaken due to an increase in the electric field in the capacitor and will completely disappear when the field in the capacitor reaches its maximum value (see Fig. 4 d).

Rice. 4. Processes occurring during one period of oscillation

When the electric field of the capacitor disappears, the magnetic field will again reach its maximum (see Fig. 4g).

The capacitor will begin charging due to the induction current. As the charge progresses, the current will weaken, and with it the magnetic field (see Fig. 4 h).

When the capacitor is charged, the current in the circuit and the magnetic field will disappear. The system will return to its original state (see Fig. 4 e).

Thus, we examined the processes occurring during one period of oscillation.

The value of energy concentrated in the electric field of the capacitor at the initial moment of time is calculated by the formula:

, Where

Capacitor charge; C- electrical capacity of the capacitor.

After a quarter of the period, all the energy of the electric field of the capacitor is converted into the energy of the magnetic field of the coil, which is determined by the formula:

Where L- coil inductance, I- current strength.

For an arbitrary moment in time, the sum of the energies of the electric field of the capacitor and the magnetic field of the coil is a constant value (if attenuation is neglected):

According to the law of conservation of energy, the total energy of the circuit remains constant, therefore, the derivative of a constant value with respect to time will be equal to zero:

Calculating derivatives with respect to time, we obtain:

Let us take into account that the instantaneous value of the current is the first derivative of the charge with respect to time:

Hence:

If the instantaneous value of the current is the first derivative of the charge with respect to time, then the derivative of the current with respect to time will be the second derivative of the charge with respect to time:

Hence:

We have obtained a differential equation whose solution is a harmonic function (the charge depends harmonically on time):

Cyclic oscillation frequency, which is determined by the values ​​of the electrical capacitance of the capacitor and the inductance of the coil:

Therefore, the oscillations of the charge, and therefore the current and voltage in the circuit, will be harmonic.

Since the oscillation period is related to the cyclic frequency by an inverse relationship, the period is equal to:

This expression is called Thomson's formula.

Bibliography

1. Myakishev G.Ya. Physics: Textbook. for 11th grade general education institutions. - M.: Education, 2010.
2. Kasyanov V.A. Physics. 11th grade: Educational. for general education institutions. - M.: Bustard, 2005.
3. Gendenstein L.E., Dick Yu.I., Physics 11. - M.: Mnemosyne

1. Lms.licbb.spb.ru ().
2. Home-task.com ().
3. Sch130.ru ().
4. Youtube.com().

Homework

1. What are electromagnetic oscillations called?
2. Questions at the end of paragraph 28, 30 (2) - Myakishev G.Ya. Physics 11 (see list of recommended readings) ().
3. How is energy converted in the circuit?

– an electrical circuit consisting of a capacitor connected in series with a capacitance, a coil with an inductance, and an electrical resistance.

Ideal oscillatory circuit- a circuit consisting only of an inductor (without its own resistance) and a capacitor (-circuit). Then, in such a system, undamped electromagnetic oscillations of the current in the circuit, the voltage on the capacitor and the charge of the capacitor are maintained. Let's look at the circuit and think about where the vibrations come from. Let an initially charged capacitor be placed in the circuit we are describing.

Rice. 1. Oscillatory circuit

At the initial moment of time, all the charge is concentrated on the capacitor, there is no current on the coil (Fig. 1.1). Because There is also no external field on the plates of the capacitor, then electrons from the plates begin to “leave” into the circuit (the charge on the capacitor begins to decrease). At the same time (due to the released electrons) the current in the circuit increases. The direction of the current, in this case, is from plus to minus (however, as always), and the capacitor represents the source alternating current for this system. However, as the current in the coil increases, as a result of , a reverse induction current () occurs. The direction of the induction current, according to Lenz's rule, should level (reduce) the increase in the main current. When the capacitor charge becomes zero (the entire charge drains), the strength of the induction current in the coil will become maximum (Fig. 1.2).

However, the current charge in the circuit cannot disappear (the law of conservation of charge), then this charge, which left one plate through the circuit, ended up on the other plate. Thus, the capacitor is recharged in the opposite direction (Fig. 1.3). The induction current on the coil decreases to zero, because the change in magnetic flux also tends to zero.

When the capacitor is fully charged, electrons begin to move in the opposite direction, i.e. the capacitor discharges in the opposite direction and a current arises, reaching its maximum when the capacitor is completely discharged (Fig. 1.4).

Further reverse charging of the capacitor brings the system to the position in Figure 1.1. This behavior of the system is repeated indefinitely. Thus, we get fluctuations in various parameters of the system: current in the coil, charge on the capacitor, voltage on the capacitor. If the circuit and wires are ideal (no intrinsic resistance), these oscillations are .

For a mathematical description of these parameters of this system (primarily, the period of electromagnetic oscillations), we introduce the previously calculated Thomson's formula:

Imperfect contour is still the same ideal circuit that we considered, with one small inclusion: with the presence of resistance (-circuit). This resistance can be either the resistance of the coil (it is not ideal) or the resistance of the conducting wires. The general logic of the occurrence of oscillations in a non-ideal circuit is similar to that in an ideal one. The only difference is in the vibrations themselves. If there is resistance, part of the energy will be dissipated into the environment - the resistance will heat up, then the energy of the oscillatory circuit will decrease and the oscillations themselves will become fading.

To work with circuits at school, only general energy logic is used. In this case, we assume that the total energy of the system is initially concentrated on and/or , and is described by:

For an ideal circuit, the total energy of the system remains constant.