Tomsono virpesių formulė statusas T sritis fizika atitikmenys: angl. Thomson's formula vok. Thomsonsche Schwingungsformel, f rus. Thomson's formula, f pranc. formule de Thomson, f ... Fizikos terminų žodynas
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- [in the name of the English. physics W. Thomson (W. Thomson; 1824 1907)] f la, expressing the dependence of the period T of sustained natural oscillations in an oscillatory circuit on its parameters of inductance L and capacitance C: T = 2PI root of LC (here L in H, C in F ... Big Encyclopedic Polytechnic Dictionary
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Expression for differenti. cross sections ds for scattering of a photon by an electron (see Compton effect). To the lab. coordinate system where the frequencies of the incident and scattered photon, the solid angle element for the scattered photon, the scattering angle, the parameter r0 = e ... Physical encyclopedia
- (Thomson) (in 1892 received the title of Baron Kelvin, Kelvin for scientific merits) (1824 1907), English physicist, member (1851) and president (1890 1895) of the Royal Society of London, foreign correspondent member (1877) and foreign honorary member ... ... encyclopedic Dictionary
- (Thomson, William), Lord Kelvin (1824 1907), English physicist, one of the founders of thermodynamics. Born in Belfast (Ireland) on June 26, 1824. He began to attend lectures of his father, professor of mathematics at the University of Glasgow, at the age of 8, and at 10 he began ... ... Collier's Encyclopedia
I Thomson Alexander Ivanovich, Russian Soviet linguist, corresponding member of the Petersburg Academy of Sciences (1910). Graduated from St. Petersburg University (1882). Professor of Novorossiysk University ...
Thomson, Lord Kelvin William (26.6.1824, Belfast, - 17.12.1907, Largs, near Glasgow; buried in London), English physicist, one of the founders of thermodynamics and kinetic theory of gases, member of the Royal Society of London ( With … Great Soviet Encyclopedia
- (Thomson, Joseph John) (1856 1940), English physicist, awarded the 1906 Nobel Prize in Physics for the work that led to the discovery of the electron. Born December 18, 1856 in the suburb of Manchester, Cheetham Hill. At the age of 14 he entered Owens ... ... Collier's Encyclopedia
Thomson's formula named after the English physicist William Thomson, who brought it out in 1853, and connects the period of natural electrical or electromagnetic oscillations in the circuit with its capacity and inductance.Thomson's formula is as follows:
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Excerpt from Thomson's Formula
- Yes, yes, I know. Let's go, let's go ... - said Pierre and entered the house. A tall, bald old man in a dressing gown, with a red nose, in galoshes on bare feet, stood in the hallway; seeing Pierre, he muttered something angrily and went into the corridor.“We had a great mind, but now, as you will see, we have grown weaker,” said Gerasim. - Would you like to go to the office? - Pierre nodded his head. - The cabinet was sealed and remained. Sofya Danilovna ordered, if they come from you, then release the books.
Pierre entered the very gloomy study into which he had entered with such trepidation during the life of the benefactor. This office, now dusty and untouched since the death of Joseph Alekseevich, was even gloomier.
Gerasim opened one shutter and tiptoed out of the room. Pierre walked around the study, went to the closet in which the manuscripts were lying, and took out one of the most important relics of the order. These were genuine Scottish acts, with notes and explanations from the benefactor. He sat down at a dusty writing table and laid the manuscripts in front of him, opened them, closed them, and finally, pushing them away from him, leaning his head on his hands, he thought.
An electromagnetic field can also exist in the absence of electric charges or currents: it is these "self-sustaining" electric and magnetic fields that are electromagnetic waves, which include visible light, infrared, ultraviolet and X-rays, 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). As in a mechanical oscillator, for example, 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 vibrations. For an oscillatory circuit, the analog 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 the capacitor. An analogue of the kinetic energy of a moving body is the energy of the 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 a 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:
Energy of vibrations. Similarly, the total electromagnetic energy of the 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 to the spring oscillator in the oscillatory circuit is the inverse capacitance of the capacitor C, and the analogue of mass is the inductance of the coil
Recall that in a mechanical system, the energy of which is given by expression (1), natural undamped harmonic oscillations can occur. The square of the frequency of such vibrations is equal to the ratio of the coefficients at the squares of the displacement and the speed in the expression for energy:
Natural frequency. In an oscillatory circuit, the electromagnetic energy of which is given by expression (2), natural 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 for the squares of the charge and the current strength):
From (4) follows an expression for the oscillation period, called the Thomson formula:
With mechanical vibrations, the dependence of the displacement x on time is determined by a cosine function, the argument of which is called the phase of the oscillations:
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 charge oscillations and the initial phase a, as in the case of a mechanical oscillator, is determined
the initial conditions, that is, the values of the capacitor charge 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 excitation conditions.
Energy transformations. Let us consider in more detail the energy transformations during mechanical and electromagnetic vibrations. In fig. 158 schematically depicts the states of mechanical and electromagnetic oscillators at 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 type to another and vice versa. The total energy of the oscillatory circuit, as well as the total energy of a mechanical oscillator, in the absence of dissipation, remains unchanged. To be convinced of this, it is necessary to substitute expression (6) for and expression for current strength in formula (2)
Using formula (4) for we obtain
Rice. 159. Graphs of the dependence on the time of the charge of the capacitor of the energy of the electric field of the capacitor and the energy of the magnetic field in the coil
The unchanged total energy coincides with the potential energy at the moments when the capacitor charge is maximum, and coincides with the magnetic field energy of the coil - the "kinetic" energy - at the moments when the capacitor charge vanishes and the current is maximum. During mutual transformations, the two types of energy perform harmonic oscillations with the same amplitude in antiphase with each other and with a frequency relative to their average value. It is easy to see this as from Fig. 158, and using the formulas of trigonometric functions of half argument:
The graphs of the time dependence of the charge of the capacitor of the energy of the electric field and the energy of the magnetic field are shown in Fig. 159 for the initial phase
The quantitative laws of natural electromagnetic oscillations can be established directly on the basis of the laws for quasi-stationary currents, without referring to the analogy with mechanical oscillations.
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 the capacitor in such a closed series circuit is equal to zero:
The voltage across the capacitor is related to the charge of the plate and to the capacitance C by the ratio The voltage across the inductance at any moment of time is equal in magnitude and opposite in the sign of the EMF of self-induction, therefore, the current in the circuit is equal to the rate of change in the charge of the capacitor: Substituting the current strength in the expression for the voltage on the inductor and denoting the second derivative of the capacitor charge with respect to time through
We obtain Now expression (10) takes the form
Let's rewrite this equation differently, introducing by definition:
Equation (12) coincides with the equation of harmonic vibrations of a mechanical oscillator with a natural frequency.The solution of such an equation is given by a harmonic (sinusoidal) function of time (6) with arbitrary values of the amplitude and initial phase a. From here follow all the above results concerning electromagnetic oscillations in the circuit.
Attenuation of electromagnetic oscillations. So far, natural oscillations in an idealized mechanical system and an idealized LC circuit have been discussed. Idealization consisted in neglecting friction in the oscillator and electrical resistance in the circuit. Only in this case will the system be conservative and the vibration energy will be conserved.
Rice. 160. Oscillatory circuit with resistance
The dissipation of the vibration energy in the circuit can be taken into account in the same way as 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 considered as an independent element in the electrical circuit of the oscillatory circuit, considering the coil and wires to be ideal (Fig. 160). When considering a quasi-stationary current in such a circuit, the voltage across the resistance must be added to equation (10)
Substituting in we get
Introducing the notation
we rewrite equation (14) as
Equation (16) for has exactly the same form as the equation for for oscillations 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 mechanical oscillations of an oscillator with viscous friction.
Dissipation of vibration energy. As in the case of mechanical vibrations, it is possible to establish a law of decay with time of the energy of natural vibrations, using the Joule-Lenz law to calculate the released heat:
As a result, in the case of small damping for time intervals much longer than the oscillation period, the rate of decrease of the oscillation energy turns out to be proportional to the energy itself:
The solution to equation (18) has the form
The energy of natural electromagnetic oscillations in a circuit with resistance decreases exponentially.
The vibration energy 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
The lifetime of the vibrations. As can be seen from (20), the amplitude of the oscillations decreases by a factor of times in a time equal 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 level of thermal noise in a given circuit. Therefore, in fact, the oscillations in the circuit "live" for a finite time, which, however, can be several times longer than the lifetime x introduced above.
It is often important to know not by itself the lifetime of the oscillations x, but the number of total 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 damping, we can conditionally speak of a period, which is understood as the time interval between two
consecutive maximum capacitor charge values (of the same polarity), or maximum current values (one direction).
Damping of oscillations affects the period, leading to its increase in comparison with the idealized case of no damping. At low damping, the increase in the oscillation period is very insignificant. However, with a strong damping, there may be no oscillations at all: the charged capacitor will discharge aperiodically, i.e., without changing the direction of the current in the circuit. This will be the case for, i.e., for
Exact solution. The laws of damped oscillations formulated above follow from the exact solution of the 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. At low damping, the factor at the cosine can be considered as a slowly varying amplitude of oscillations.
Task
Recharging capacitors through the 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. At the initial moment of time, only one capacitor is charged
Rice. 162. Charges of capacitors and current in the circuit after the key is closed
Rice. 163. Mechanical analogy for the electrical circuit shown in fig. 162
Solution. After the key is closed, oscillations arise in the circuit: the upper capacitor begins to discharge through the coil, while charging the lower one; then everything happens in the opposite direction. Suppose, for example, that the upper plate of the capacitor is positively charged at. Then
after a short time interval, the signs of the charges of the capacitor plates and the direction of the current will be as shown in Fig. 162. Let's designate by the charges of those plates of the upper and lower capacitors, which are connected to each other through the inductor. Based on the law of conservation of electric charge
The sum of the stresses on all elements of the closed loop at each moment of time is equal to zero:
The sign of the voltage across the capacitor corresponds to the charge distribution in Fig. 162. and the indicated direction of the current. The expression for the current through the coil can be written in either of two forms:
We 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 sustained 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, we take into account that at the initial moment the charge a is the current For the current strength from (24) and (31) we have
Since it follows that Substituting now into and taking into account that we obtain
So, the expressions for the charge and the current are of the form
The nature of the charge and current oscillations is especially evident at the same values of the capacitance of the capacitors. In this case
The charge of the upper capacitor oscillates with an amplitude about an average value equal to. For half the oscillation period, it decreases from the maximum value at the initial moment to zero, when all the 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 entering the circuit with one equivalent.
A visual representation of the processes taking place here is given by a mechanical analogue of this electrical circuit, shown in Fig. 163. Identical springs correspond to the case of capacitors 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 here 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 undeformed, and the right one is compressed in the same way as the left one at the initial moment, which corresponds to a complete overflow of the charge from one capacitor to another. Although the ball performs normal harmonic vibrations 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 one capacitor, where during oscillations its repeated full recharging occurs, in the considered system the initially charged capacitor is not completely recharged. For example, when its charge decreases to zero and then recovers again in the same polarity. Otherwise, these vibrations do not differ from harmonic vibrations in a conventional circuit. The energy of these vibrations 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 analogue of stiffness k is and the analogue of mass inductance and not vice versa.
Give the 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 -contour are characterized by amplitude, frequency, period, phase of oscillations, initial phase. Which of these quantities are determined by the properties of the oscillatory circuit itself, and which depend on the method of excitation of oscillations?
Prove that the average values of the 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 the oscillations.
How to apply the laws of quasi-stationary phenomena in an electric circuit to derive the differential equation (12) of harmonic oscillations in a -loop?
What differential equation does the current in the LC circuit satisfy?
Conduct the derivation of the equation for the rate of decrease in the energy of oscillations at low damping in the same way as it was done for a mechanical oscillator with friction proportional to the velocity, and show that for time intervals significantly exceeding the period of oscillations, this decrease occurs exponentially. What is the meaning of the term "low attenuation" used here?
Show that the function given by formula (21) satisfies equation (16) for any values and a.
Consider the mechanical system shown in fig. 163, and find the time dependence of the deformation of the left spring and the velocity of a massive body.
A loop without resistance with inevitable losses. In the problem considered above, despite the not quite ordinary initial conditions for charges on capacitors, it was possible to apply the usual equations for electric circuits, since the conditions for 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 similarity to the circuit in Fig. 162, the conditions of quasi-stationarity are not met if at the initial moment one capacitor is charged and the other is not.
Let us discuss in more detail the reasons why the quasi-stationarity conditions are violated here. Immediately after closing
Rice. 164. An electric circuit for which the conditions of quasi-stationarity are not met
key, all processes are played out only in capacitors connected to each other, 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 short, since the inductance of the connecting wires is low. 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 together with the redistribution of charges, a redistribution of energy also occurs, part of which passes into heat.
After damping of fast oscillations, oscillations occur in the system, as in a circuit with one 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 in comparison with the inductance of the coil.
As in the considered problem, it is useful to find a mechanical analogy here as well. If there two springs corresponding to the capacitors were located on both sides of a massive body, then here they should be located on one side of it, so that the vibrations of one of them could be transmitted to the other when the body is stationary. Instead of two springs, you can take one, but only at the initial moment it should be deformed non-uniformly.
We grab the spring by the middle and stretch its left half by a certain 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. As you can easily imagine, the stiffness of the "half" of the spring is equal. If the mass of the spring is small compared to the mass of the ball, the frequency of natural vibrations of the spring as an extended system is much greater than the frequency of vibrations of the ball on the spring. These "fast" oscillations will decay in a time that is a small fraction of the period of oscillation of the ball. After the damping of fast oscillations, the tension in the spring is redistributed, and the displacement of the load practically remains the same, since the load does not have time to noticeably move during this time. The deformation of the spring becomes homogeneous, 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 uniform initial deformation. The dependence of the displacement of the load on time is expressed by the same formula (36).
In the example considered, as a result of rapid oscillations, half of the initial supply of mechanical energy turned into internal energy (heat). It is clear that by subjecting the initial deformation not to half, but to an arbitrary part of the spring, any fraction of the initial supply of mechanical energy can be converted into internal energy. But in all cases, the energy of vibrations of the load on the spring corresponds to the energy reserve with the same uniform initial deformation of the spring.
In an electrical 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 of the initial energy supply. The second 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, idealization, in which the dissipation of the vibration energy is neglected, is in principle inadmissible in this system. The reason for this is that rapid oscillations are possible here, without affecting the inductors or a massive body in a similar mechanical system.
Oscillatory circuit with nonlinear elements. When studying mechanical vibrations, we saw that vibrations are far from always harmonic. Harmonic vibrations are a characteristic property of 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 the 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, that is, independent of the charge and current. The dielectric capacitor and the core coil are, strictly speaking, non-linear elements. When a capacitor is filled with a ferroelectric, that is, a substance whose dielectric constant strongly depends on the applied electric field, the capacitance of the capacitor can no longer be considered constant. Likewise, the inductance of a coil with a ferromagnetic core depends on the strength of the current, since the ferromagnet has the property of magnetic saturation.
If in mechanical oscillatory systems, mass, as a rule, can be considered constant and nonlinearity arises only due to the nonlinear nature of the acting force, then in an electromagnetic oscillatory circuit nonlinearity can arise both due to a capacitor (analogue of an elastic spring) and due to an inductor ( analogue mass).
Why idealization, in which the system is considered conservative, is inapplicable for an oscillatory circuit with two parallel capacitors (Fig. 164)?
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-sinusoidal electromagnetic oscillations in the circuit?
- 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 give it a negative charge, the electrons leaving the neutral plate give it a positive charge. 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
- 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.
