The period of own electromagnetic oscillations of the formula. Electromagnetic oscillations

Themes of the EGE codifier: Free electromagnetic oscillations, oscillatory outline, forced electromagnetic oscillations, resonance, harmonic electromagnetic oscillations.

Electromagnetic oscillations - These are periodic changes in charge, current and voltage forces occurring in an electrical circuit. Simplest system To observe electromagnetic oscillations, the oscillating circuit is served.

Oscillating contour

Oscillating contour- This is a closed circuit formed by a contented condenser and coil.

Charge a capacitor, connect the coil to it and closed the chain. Start occurble free electromagnetic oscillations - Periodic charge changes on the condenser and current in the coil. Free, recall, these oscillations are called because they are performed without any external influence - only at the expense of energy stored in the circuit.

The period of oscillations in the circuit will indicate, as always, through. The coil resistance will be considered equal to zero.

Consider in detail all the important stages of the process of oscillations. For greater clarity, we will conduct an analogy with oscillations of the horizontal spring pendulum.

Starting:. The capacitor charge is equal to the current through the coil (Fig. 1). The condenser will now begin to discharge.

Fig. one.

Despite the fact that the resistance of the coil is zero, the current will not increase instantly. As soon as the current starts to increase, self-induction EMF will arise in the coil, which prevents the increase in current.

Analogy. The pendulum is drawn to the right on the magnitude and is released at the initial moment. The initial speed of the pendulum is zero.

The first quarter of the period:. The capacitor is discharged, its charge is currently equal. The current through the coil is growing (Fig. 2).

Fig. 2.

An increase in current occurs gradually: the vortex electric field of the coil prevents the current increases and is directed against the current.

Analogy. The pendulum moves to the left to the position of equilibrium; The speed of the pendulum gradually increases. Spring deformation (it is the coordinate of the pendulum) decreases.

End of the first quarter:. The condenser completely discharged. The strength of the current reached the maximum value (Fig. 3). The condenser recharge will now begin.

Fig. 3.

The voltage on the coil is zero, but the current will not disappear instantly. As soon as the current starts decreasing, self-induction EMF will appear in the coil, which prevents decreasing current.

Analogy. The pendulum is the position of the equilibrium. Its speed reaches the maximum value. Spring deformation is zero.

Two quarter:. The capacitor is recharged - a charge of the opposite sign appears on its plates compared to what was at first (Fig. 4).

Fig. four.

The strength of the current decreases gradually: the vortex electric field of the coil, maintaining the decreases current, is coated with a current.

Analogy. The pendulum continues to move left - from the equilibrium position to the rightmost point. His speed gradually decreases, the springs deformation increases.

End of the second quarter . The condenser completely recharged, its charge is equal (but the polarity is different). The current is zero (Fig. 5). The condenser reverse recharge will now begin.

Fig. five.

Analogy. The pendulum has reached the extreme right point. The speed of the pendulum is zero. Spring deformation is maximum and equal.

Third quarter:. The second half of the oscillation period began; The processes went in the opposite direction. The capacitor is discharged (Fig. 6).

Fig. 6.

Analogy. The pendulum moves back: from the rightmost point to the position of equilibrium.

The end of the third quarter:. The condenser completely discharged. The current is maximum and is equal again, but this time has a different direction (Fig. 7).

Fig. 7.

Analogy. The pendulum again passes the position of equilibrium at maximum speed, but this time in the opposite direction.

Fourth quarter:. The current decreases, the capacitor is charging (Fig. 8).

Fig. eight.

Analogy. The pendulum continues to move right - from the position of equilibrium to the extreme left point.

The end of the fourth quarter and the whole period:. Reverse recharge of the condenser is complete, the current is zero (Fig. 9).

Fig. nine.

This moment is identical to the moment, and this drawing is Figure 1. One complete oscillation was made. The following oscillation will begin, during which the processes will occur in the same way as described above.

Analogy. The pendulum returned to its original position.

Considered electromagnetic oscillations are unlucky - They will continue indefinitely. After all, we suggested that the coil resistance is zero!

In the same way, there will be unlucky fluctuations in the spring pendulum in the absence of friction.

In reality, the coil has some resistance. Therefore, fluctuations in the real oscillatory circuit will be attenuating. So, after one complete oscillation of the charge on the condenser will be less than the source value. Over time, the oscillations will be disappeared at all: all the energy, well-stained in the circuit, is highlighted in the form of heat on the resistance of the coil and connecting wires.

In the same way, the fluctuations in the real spring pendulum will be attenuating: all the pendulum energy will gradually turn into heat due to the inevitable presence of friction.

Energy transformations in the oscillatory circuit

We continue to consider the unlucky oscillations in the contour, considering the resistance of the coil of zero. The capacitor has a container, the inductance of the coil is equal.

Since there are no heat losses, the energy from the contour does not go away: it is constantly redistributed between the condenser and the coil.

Take the moment when the capacitor charge is maximum and is equal, and there is no current. The energy of the magnetic field of the coil at this moment is zero. All the energy of the contour is concentrated in the condenser:

Now, on the contrary, consider the moment when the current is maximum and equal, and the condenser is discharged. The energy of the capacitor is zero. All contour energy is stored in the coil:

At an arbitrary moment when the capacitor charge is equal to the current flow, the circuit energy is equal to:

In this way,

(1)

The ratio (1) is used in solving many tasks.

Electromechanical analogies

In the previous sheet of self-induction, we noted an analogy between inductance and mass. Now we can set a few more compliances between electrodynamic and mechanical values.

For a spring pendulum, we have a ratio similar to (1):

(2)

Here, as you already understood, the rigidity of the spring is the mass of the pendulum, and the current values \u200b\u200bof the coordinate and the speed of the pendulum, and are their greatest meanings.

By comparing each other equality (1) and (2), we see the following compliance:

(3)

(4)

(5)

(6)

Relying on these electromechanical analogies, we can foresee the formula for the period of electromagnetic oscillations in the oscillatory circuit.

In fact, the period of oscillations of the spring pendulum, as we know, is equal to:

B compliance with the analogies (5) and (6) replace the mass into inductance here, and the rigidity on the reverse tank. We get:

(7)

Electromechanical analogies are not supplied: Formula (7) gives a true expression for the oscillation period in the oscillatory circuit. It is called thomson formula. We will soon give it more strict output.

Harmonic oscillation law in contour

Recall that oscillations are called harmonicIf the oscillating value changes with time according to the law of sine or cosine. If you managed to forget these things, be sure to repeat the "Mechanical oscillations" leaves.

Charge fluctuations on the condenser and current strength in the circuit are harmonic. We now prove it. But former, we need to establish the rules for choosing a sign for the charge of the capacitor and for the strength of the current - after all, with oscillations, these values \u200b\u200bwill be taken both positive and negative values.

First we choose positive direction bypass contour. The choice of role does not play; Let it be a direction counterclock-wise (Fig. 10).

Fig. 10. Positive Bypass

The current is considered positive class \u003d "Tex" alt \u003d "(! Lang: (I\u003e 0)"> , если ток течёт в положительном направлении. В противном случае сила тока будет отрицательной .!}

The charge of the condenser is the charge of that of its plate, on which The positive current flows (i.e., the plates on which the arrow of the partition direction indicates). In this case, the charge leva Plates of the condenser.

With such a choice of current and charge signs, the ratio is true: (with another choice of signs it could happen). Indeed, signs of both parts coincide: if class \u003d "Tex" alt \u003d "(! Lang: I\u003e 0"> , то заряд левой пластины возрастает, и потому !} class \u003d "Tex" alt \u003d "(! Lang: \\ Dot (Q)\u003e 0"> !}.

The values \u200b\u200band change over time, but the energy of the contour remains unchanged:

(8)

It became, the energy derivative in time turns to zero :. We take the time derivative from both parts of the relation (8); Do not forget that complex functions are differentiated on the left (if - the function from, then according to the range of differentiation of a complex function, the derivative of our function is equal to :) :):

Substituting here and, we get:

But the strength of the current is not a function identically equal to zero; so

Let's rewrite it in the form:

(9)

We obtained the differential equation of harmonic oscillations of the species, where. This proves that the charge of the capacitor fluctuates the harmonic law (i.e., according to the law of sinus or cosine). The cyclic frequency of these oscillations is:

(10)

This value is called more own frequency contour; It is with this frequency in the circuit free of charge (or, as they say, own oscillations). The oscillation period is:

We came to the Thomson formula again.

Harmonic charge charges in the general case has the form:

(11)

Cyclic frequency is in formula (10); The amplitude and the initial phase are determined from the initial conditions.

We will consider the situation in detail studied at the beginning of this leaf. Suppose when the capacitor is maximum and equal to (as in Fig. 1); There is no current in the circuit. Then the initial phase, so the charge changes under the law of cosine with amplitude:

(12)

We will find the law of change of current. For this, the ratio (12) is differentiating the time, again, not forgetting the rule of the derivative of a complex function:

We see that the current of the current changes by the harmonic law, this time - according to the law of sinus:

(13)

The current amplitude is equal to:

The presence of "minus" in the law of change of current (13) is not difficult to understand. Take, for example, time interval (Fig. 2).

The current flows in the negative direction :. Since, the oscillation phase is in the first quarter :. Sinus in the first quarter is positive; Therefore, sinus in (13) will be positive on the time interval under consideration. Therefore, to ensure the negativity of the current, the minus sign in formula (13) is really necessary.

And now look in Fig. eight . The current flows in the positive direction. How does our "minus" work in this case? Understand what is the case!

I will depict the charts of charge and current oscillations, i.e. Fun graphics (12) and (13). For clarity, imagine these graphs in the same coordinate axes (Fig. 11).

Fig. 11. Charges of charge and current oscillations

Please note: zeros of charge are per maxims or current; Conversely, the current zeros correspond to maxima or minima charge.

Using the brief formula

we write the law of changing the current (13) in the form:

By comparing this expression with the law of changing the charge, we see that the current phase is equal, more charge phase by magnitude. In this case, they say that the current ahead of phase charge on; or shift phases between current and charge is equal; or phase difference Between current and charge is equal.

Ahead of the phase charge current is graphically manifested in the fact that the current schedule is shifted left On relatively chart. The strength of the current reaches, for example, a quarter of a period of a quarter earlier than the maximum of the charge reaches (and a quarter of the period just corresponds to the phase difference).

Forced electromagnetic oscillations

As you remember, forced oscillations arise in the system under the action of periodic forcing force. Frequency forced oscillations Coincides with the frequency of the forcing force.

Forced electromagnetic oscillations will be accomplished in the circuit, swamped to the source of sinusoidal voltage (Fig. 12).

Fig. 12. Forced oscillations

If the source voltage varies by law:

then the circuit takes place charge and current with cyclic frequency (and with a period, respectively,). The source of alternating voltage as it were "imposes" the contour of its oscillation frequency, causing to forget about its own frequency.

The amplitude of the forced oscillations of charge and current depends on the frequency: the amplitude is the greater, the closer to the circuit's own frequency. And comes resonance - sharp increase in the amplitude of oscillations. We will talk about resonance in more detail in the next sheet dedicated to the variable current.

The electrical circuit consisting of the inductor inductance and the condenser (see Figure) is called a oscillating circuit. In this chain there may be peculiar electrical oscillations. Let, for example, in the initial moment of time we charge the condenser plates with positive and negative charges, and then allow the charges to move. If the coil was absent, the capacitor would start to be discharged, in the chain for a short time electricity, and the charges would have gone. Here is the following. First, thanks to self-induction, the coil prevents the current increases, and then when the current begins to decrease, prevents its decrease, i.e. Supports current. As a result, self-induction EMF charges the condenser with reverse polarity: the plate that was originally charged positively, acquires a negative charge, the second is positive. If there is no loss of electrical energy (in the case of low resistance of the contour elements), the value of these charges will be the same as the value of the initial charges of the condenser plates. In the future, the process of moving charges will be repeated. Thus, the movement of charges in the circuit is an oscillatory process.

To solve the problems of EGE devoted to electromagnetic oscillations, you need to remember a number of facts and formulas relating to the oscillating circuit. First, you need to know the formula for the oscillation period in the circuit. Secondly, be able to apply the law of energy conservation to the oscillatory contour. And finally (although such tasks are rarely found), be able to use the dependence of current force through the coil and voltage on the condenser from time

The period of electromagnetic oscillations in the oscillatory circuit is determined by the ratio:

where and is the charge on the condenser and the strength of the current in the coil at this point in time, and the capacitance of the capacitor and the inductance of the coil. If the electrical resistance of the circuit elements is not enough, then the electrical energy of the circuit (24.2) remains almost unchanged, despite the fact that the capacitor charge and the current in the coil change over time. From formula (24.4) it follows that energy transformations occur during electrical oscillations in the circuit: in those moments when the current in the coil is zero, all the energy of the circuit is reduced to the energy of the condenser. In those moments of time, when equal to zero capacitor charge, the circuit energy is reduced to the magnetic field energy in the coil. Obviously, during these moments of time, the charge of the capacitor or current in the coil reaches its maximum (amplitude) values.

In electromagnetic oscillations in the circuit, the capacitor charge changes over time by harmonic law:

standard for any harmonic oscillations. Since the strength of the current in the coil is a time derivative of a time capacitor, from formula (24.4), you can find the dependence of the current in the coil from time

In physics, there are often challenges on electromagnetic waves. The minimum of knowledge necessary for solving these tasks includes an understanding of the main properties of an electromagnetic wave and knowledge of the scale of electromagnetic waves. We formulate these facts and principles.

According to the electromagnetic field laws, the alternating magnetic field generates an electric field, an alternating electric field generates a magnetic field. Therefore, if one of the fields (for example, electric) begins to change, the second field (magnetic) will occur, which then reveals the first (electric) again, then the second (magnetic), etc. The process of mutual transformation into each other electrical and magnetic fields, which can be distributed in space, is called an electromagnetic wave. Experience shows that the directions in which the vectors of the electric and induction vectors of the magnetic field fluctuate the electromagnetic wave perpendicular to the direction of its propagation. This means that electromagnetic waves are transverse. In the theory of the electromagnetic field, Maxwell proves that the electromagnetic wave is created (emitted) electric charges When they are moving with acceleration. In particular, the source of the electromagnetic wave is the oscillating circuit.

The length of the electromagnetic wave, its frequency (or period) and the rate of propagation are associated with the relation, which is valid for any wave (see also formula (11.6)):

Electromagnetic waves in vacuum apply at speeds \u003d 3 10 8 m / s, in the medium the speed of electromagnetic waves is less than in vacuum, and this speed depends on the frequency of the wave. Such a phenomenon is called dispersion of waves. The electromagnetic wave is inherent in all the properties of waves propagating in elastic media: interference, diffraction, the Guygens principle is valid for it. The only thing that distinguishes the electromagnetic wave is that it does not need an environment for its distribution - an electromagnetic wave can be distributed in vacuo.

In nature, electromagnetic waves are observed with frequencies heavily different from each other, and have significantly different properties (despite the same physical nature). The classification of the properties of electromagnetic waves depending on their frequency (or wavelength) is called the scale of electromagnetic waves. Let us give a brief overview of this scale.

Electromagnetic waves with a frequency of less than 10 5 Hz (that is, with a wavelength, more than several kilometers) are called low-frequency electromagnetic waves. Most household electrical devices emit such a range.

Waves with a frequency of 10 5 to 10 12 Hz are called radio waves. These waves correspond to wavelengths in vacuum from several kilometers to a few millimeters. These waves are used for radio communications, television, radar, cell phones. Sources of radiation of such waves are charged particles moving in electromagnetic fields. Radio waves are also emitted by free metal electrons that make oscillations in the oscillatory circuit.

The scope of electromagnetic waves with frequencies lying in the range of 10 12 - 4.3 10 14 Hz (and wavelengths from several millimeters to 760 nm) is called infrared radiation (or infrared beams). The source of such radiation serve the molecules of the heated substance. A person radiates infrared waves with a wavelength of 5 - 10 microns.

Electromagnetic radiation in the frequency range 4.3 10 14 - 7.7 10 14 Hz (or wavelength 760 - 390 nm) is perceived by the human eye as light and is called visible light. Waves of different frequencies within this range are perceived by the eye, as having a different color. The wave with the smallest frequency of the visible range of 4.3 10 14 is perceived as red, with the largest frequency inside the visible range of 7.7 10 14 Hz - as purple. Visible light is radiated when moving electrons in atoms, solid tel molecules heated to 1000 ° C and more.

Waves with a frequency of 7.7 10 14 - 10 17 Hz (wavelength from 390 to 1 nm) It is customary to be called ultraviolet radiation. Ultraviolet radiation has a pronounced biological effect: it is able to kill a number of microorganisms, it is capable of increasing the pigmentation of human skin (tan), in excess irradiation in some cases, can contribute to the development of cancer (skin cancer). Ultraviolet rays are contained in the radiation of the Sun, in laboratories are created by special gas-discharge (quartz) lamps.

Behind the area of \u200b\u200bultraviolet radiation is the region of the X-ray rays (frequency 10 17 - 10 19 Hz, the wavelength from 1 to 0.01 nm). These waves are emitted when braking in the substance of charged particles, overclocked with a voltage of 1000 V and more. Possess the ability to pass through thick layers of the substance, opaque for visible light or ultraviolet radiation. Thanks to this property, X-rays are widely used in medicine to diagnose bone fractures and a number of diseases. X-rays have a destructive effect on biological tissue. Due to this property, they can be used to treat oncological diseases, although in excess radiation they are deadly dangerous for a person, causing a number of violations in the body. Due to the very low wavelength, the wave properties of X-ray radiation (interference and diffraction) can be detected only on structures comparable to the size of atoms.

Gamma radiation (-Exusual) is called electromagnetic waves with a frequency, greater than 10 20 Hz (or wavelength, less than 0.01 nm). There are such waves in nuclear processes. A feature of the emission is its pronounced corpuscular properties (that is, this radiation behaves like a flow of particles). Therefore, about-emission is often spoken as a stream of -chasts.

IN task 24.1.1 To establish conformity between units of measurements, we use formula (24.1), from which it follows that the period of oscillations in the circuit with a capacitor with a capacity of 1 F and an inductance of 1 GG is equal to seconds (response 1 ).

From the schedule given to task 24.1.2., we conclude that the period of electromagnetic oscillations in the circuit is 4 ms (answer 3 ).

According to the formula (24.1) we find the period of oscillations in the circuit, given in task 24.1.3.:
(answer 4 ). It should be noted that according to the scale of electromagnetic waves of such a circuit, the waves of a long-wave radio view is emitted.

The period of oscillation is called the time of one complete oscillation. This means that if at the initial moment of time the capacitor is charged with a maximum charge ( task 24.1.4.), after half a period, the capacitor will also be charged with a maximum charge, but with reverse polarity (the plate that was originally charged positively, will be negatively charged). And the maximum current in the circuit will be achieved between these two moments, i.e. Through a quarter of the period (answer 2 ).

If you increase the inductance of the coil four times ( task 24.1.5.), according to the formula (24.1), the period of oscillations in the contour will increase twice, and the frequency decrease twice (answer 2 ).

According to formula (24.1), with an increase in the capacitance of the capacitor, four times ( task 24.1.6) The period of oscillations in the circuit increases twice (answer 1 ).

When closing the key ( task 24.1.7) In the circuit, two of the same capacitor connected in parallel (see Figure) will operate instead of one capacitor. And since when parallel compound The capacitors of their capacity are folded, the closure of the key leads to a two-time increase in the circuit of the contour. Therefore, from formula (24.1) we conclude that the period of oscillations increases at once (answer 3 ).

Let the charge on the condenser makes oscillations with a cyclic frequency ( task 24.1.8.). Then, according to formulas (24.3) - (24.5), with the same frequency, fluctuations in the current in the coil will be performed. This means that the current dependence on time can be represented as . From here we find the dependence of the energy of the magnetic field of the coil from time

From this formula it follows that the magnetic field energy in the coil makes oscillations with a double frequency, and, it means, with a period, twice as smaller period of charge and current oscillation period (answer 1 ).

IN task 24.1.9 We use the law of conservation of energy for the oscillating circuit. From formula (24.2) it follows that for amplitude voltage values \u200b\u200bon the condenser and current in the coil, the ratio is fair

where and the amplitude values \u200b\u200bof the charge of the capacitor and the current in the coil. From this formula using relation (24.1) for the oscillation period in the circuit we find an amplitude current

answer 3 .

Radio waves - electromagnetic waves with certain frequencies. Therefore, the speed of their propagation in vacuo is equal to the rate of propagation of any electromagnetic waves, and in particular X-ray. This speed is the speed of light ( task 24.2.1 - answer 1 ).

As mentioned earlier, the charged particles emit electromagnetic waves when moving with acceleration. Therefore, the wave is not radiated only with uniform and rectilinear movement ( task 24.2.2. - answer 1 ).

An electromagnetic wave is a specially changing in space and time and supporting each other electrical and magnetic fields. Therefore, the correct answer in task 24.2.3. - 2 .

From given to the condition tasks 24.2.4. Graphics follows that the period of this wave is \u003d 4 μs. Therefore, from formula (24.6) we get M (answer 1 ).

IN task 24.2.5. By formula (24.6) we find

(answer 4 ).

The oscillating circuit is associated with an antenna receiver of electromagnetic waves. The electric field of the wave acts on the free electrons in the circuit and makes them perform oscillations. If the frequency of the wave coincides with its own frequency of electromagnetic oscillations, the amplitude of oscillations in the circuit increases (resonance) and can be registered. Therefore, to receive electromagnetic waves, the frequency of own oscillations in the circuit must be close to the frequency of this wave (the circuit must be configured to the frequency of the wave). Therefore, if the contour needs to be reconfigured from a wave with a length of 100 m on a wave of 25 m long ( task 24.2.6), the own frequency of electromagnetic oscillations in the circuit must be increased by 4 times. For this, according to formulas (24.1), (24.4) the capacitance capacitor should be reduced 16 times (answer 4 ).

According to the scale of electromagnetic waves (see the introduction to this chapter), maximum length from listed in the condition tasks 24.2.7 Electromagnetic waves have the radiation of the radio transmitter antenna (answer 4 ).

Among the listed B. task 24.2.8 Electromagnetic waves with a maximum frequency of X-ray radiation (answer 2 ).

The electromagnetic wave is transverse. This means that the vectors of the electric field and the induction of the magnetic field in the wave at any time are directed perpendicular to the direction of the wave propagation. Therefore, when the wave is propagated in the axis direction ( task 24.2.9), the electric field strength vector is directed perpendicular to this axis. Consequently, it is always equal to zero its projection on the axis \u003d 0 (answer 3 ).

The speed of propagation of the electromagnetic wave is the individual characteristic of each medium. Therefore, when moving an electromagnetic wave from one medium to another (or from a vacuum to Wednesday), the speed of the electromagnetic wave changes. And what can be said about two other wavelengths included in formula (24.6), the wave and frequency length. Will they change when the wave transition from one environment to another ( task 24.2.10)? Obviously, the frequency of the wave does not change when moving from one environment to another. Indeed, the wave is an oscillatory process in which an alternating electromagnetic field in one environment creates and supports the field in another environment due to precisely these changes. Therefore, the periods of these periodic processes (and therefore frequencies) in one and the other environment should coincide (answer 3 ). And since the speed of the wave in different environments is different, then from the arguments and formulas (24.6) it follows that the wavelength during its transition from one medium to another - changes.

The main device determining the operating frequency of any generator alternating current, is the oscillating circuit. The oscillating circuit (Fig. 1) consists of a coil inductance L. (Consider the ideal case when the coil does not have ohmic resistance) and the condenser C. And called closed. The characteristic of the coil is inductance, it is indicated L. and is measured in Henry (GG), the capacitor is characterized by a container C.which is measured in the Farades (f).

Let the capacitor are charged at the initial moment of time (Fig. 1) that on one of his plates there is a charge + Q. 0, and on the other - charge - Q. 0. At the same time, an electric field with energy is formed between the capacitor plates.

where - amplitude (maximum) voltage or potential difference on capacitor plates.

After the circuit circuit, the capacitor begins to discharge and the circuit will go the electric current (Fig. 2), the value of which increases from zero to the maximum value. Since a variable current flows in the chain, then self-induction EMP is induced in the coil, which prevents the discharge of the capacitor. Therefore, the process of dischargeing the condenser does not occur instantly, but gradually. At every moment of time, the potential difference on the capacitor plates

(where - the charge of the condenser at the moment time) is equal to the potential difference on the coil, i.e. Equal to EMF self-induction

Fig.1	Fig.2

When the condenser is completely discharged and, the current strength in the coil reaches the maximum value (Fig. 3). The induction of the magnetic field of the coil at this moment is also maximum, and the magnetic field energy will be equal to

The current of the current begins to decrease, and the charge will accumulate on the condenser plates (Fig. 4). When the current is reduced to zero, the capacitor charge reaches the maximum value Q. 0, but the label, first charged positively, will now be charged negatively (Fig. 5). The condenser is then again begins to discharge, and the current in the chain flows in the opposite direction.

So the process of flowing the charge from one condenser clamping to another through the inductor is repeated again and again. They say that the circuit occurs electromagnetic oscillations . This process is associated not only with the oscillations of the charge value and voltage on the condenser, the current forces in the coil, but also the pumping of the energy from the electric field to magnetic and back.

Fig. 3.	Fig.4.

Recharge the capacitor to the maximum voltage will occur only if there is no energy loss in the oscillatory circuit. Such contour is called perfect.

In real circuits, the following energy loss takes place:

1) thermal losses, because R. ¹ 0;

2) losses in the dielectric condenser;

3) hysteresis losses in the core coil;

4) Losses on radiation, etc. If you neglect by these losses of energy, then you can write that, i.e.

Oscillations occurring in the perfect oscillatory circuit in which this condition is being done are called free, or own, oscillations of the contour.

In this case voltage U. (and charge Q.) The capacitor varies on the harmonic law:

where n is the intrinsic frequency of the oscillating circuit, W 0 \u003d 2pn - its own (circular) frequency of the oscillating circuit. The frequency of electromagnetic oscillations in the circuit is defined as

T. - time during which one complete voltage fluctuation on the condenser and current in the circuit is performed, is determined Thomson formula

The strength of the current in the circuit also changes in harmonic law, but lags behind the voltage of phase. Therefore, the dependence of the current in the circuit will be viewed

. (9)

Figure 6 presents voltage change graphs U. on the condenser and current I. In the coil for an ideal oscillatory circuit.

In the real circuit, energy with each oscillation will decrease. The amplitudes of the voltage on the condenser and current in the circuit will decrease, such oscillations are called decaying. In the specifying generators it is impossible to apply them because The device will work at best in pulse mode.

Fig.5	Fig.6.

To obtain unlucky oscillations, it is necessary to compensate for the loss of energy with a wide variety of operating frequencies, including those used in medicine.

• Electromagnetic oscillations - These are periodic changes with the time of electrical and magnetic values \u200b\u200bin the electrical circuit.

• Free are called such oscillationswhich occur in a closed system due to the deviation of this system on the state of steady equilibrium.

With oscillations, there is a continuous process of converting the energy of the system from one form to another. In the case of oscillations of the electromagnetic field, the exchange can only go between the electrical and magnetic component of this field. The simplest system where this process can occur is oscillating contour.

• Perfect oscillating contour (LC-contour) - electrical circuitconsisting of coil inductance L. and capacitor capacity C..

Unlike a real oscillatory circuit, which has electrical resistance R.The electrical resistance of the ideal contour is always equal to zero. Consequently, the perfect oscillating circuit is a simplified model of a real circuit.

Figure 1 shows the scheme of an ideal oscillatory circuit.

Energy contour

Complete energy of the oscillating circuit

\\ (W \u003d w_ (e) + w_ (m), \\; \\; \\; w_ (E) \u003d \\ DFRAC (C \\ CDOT U ^ (2)) (2) \u003d \\ DFRAC (Q ^ (2)) (2c), \\; \\; \\; w_ (m) \u003d \\ dfrac (L \\ Cdot i ^ (2)) (2), \\)

Where W E. - the energy of the electrical field of the oscillating circuit at the moment, FROM - electrical capacity of the capacitor, u. - voltage value on the condenser at a given time, q. - the value of the capacitor's charge at the moment, W M. - the energy of the magnetic field of the oscillating circuit at the moment, L. - inductance of the coil, i. - Current strength in the coil at the moment.

Processes in the oscillatory circuit

Consider the processes that occur in the oscillatory circuit.

To remove the contour from the equilibrium position charge the capacitor so that on its plates will be charged Q M. (Fig. 2, position 1 ). Taking into account the equation \\ (u_ (m) \u003d \\ dfrac (q_ (m)) (C) \\) we find the voltage value on the condenser. Current in the chain at this point in time is not, i.e. i. = 0.

After closing the key under the action of the electrical field of the capacitor in the circuit, an electric current will appear, the current i. which will increase over time. The capacitor will start discharge at this time, because Electrons, creating a current, (remind you that the direction of current is taken by the direction of the movement of positive charges) leave with a negative condenser clamp and come to positive (see Fig. 2, position 2 ). Together with the charge q. will decrease and voltage u. \\ (\\ left (u \u003d \\ dfrac (q) (C) \\ RIGHT). \\) With an increase in the current strength through the coil, self-induction will arise, which prevents the change in the current strength. As a result, the current of the current in the oscillatory circuit will increase from zero to some maximum value not instantly, but for a certain period of time determined by the inductance of the coil.

Capacitor charge q. decreases and at some point in time it becomes zero ( q. = 0, u. \u003d 0), the current of the current in the coil will reach some value I M. (see Fig. 2, position 3 ).

Without the electric field of the condenser (and resistance), the electrons that create the current continue their inertia movement. At the same time, electrons coming to a neutral capacitor clamp report to it a negative charge, electrons leaving neutrally inform her positive charge. On the condenser begins to appear q. (and voltage u.), but the opposite sign, i.e. Condenser recharges. Now the new electrical field of the capacitor prevents the electron movement, so the current i. starts decree (see Fig. 2, position 4 ). Again, this does not happen instantly, since now EMF self-induction seeks to compensate for the decrease in current and "supports" it. And the value of the current I M. (pregnant 3 ) It turns out maximum current value in the circuit.

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

As the charge on the condenser q. (and voltage u.) Determines its electric field energy W E. \\ (\\ left (w_ (e) \u003d \\ dfrac (q ^ (2)) (2c) \u003d \\ DFRAC (C \\ CDOT U ^ (2)) (2) \\ Right), \\) and current power in the coil i. - magnetic field energy WM. \\ (\\ left (W_ (M) \u003d \\ DFRAC (L \\ CDOT I ^ (2)) (2) \\ Right), \\) then, together with changes in charge, voltage and current, will change and energy.

Designation in the table:

\\ (W_ (e \\, \\ max) \u003d \\ dfrac (q_ (m) ^ (2)) (2c) \u003d \\ dfrac (C \\ CDOT U_ (M) ^ (2)) (2), \\; \\; \\; W_ (e \\, 2) \u003d \\ dfrac (q_ (2) ^ (2)) (2c) \u003d \\ DFRAC (C \\ CDOT U_ (2) ^ (2)) (2), \\; \\; \\ W_ (E \\, 6) \u003d \\ DFRAC (Q_ (6) ^ (2)) (2C) \u003d \\ DFRAC (C \\ CDOT U_ (6) ^ (2)) (2), \\)

\\ (W_ (m \\; \\ max) \u003d \\ dfrac (l \\ cdot i_ (m) ^ (2)) (2), \\; \\; \\; w_ (m2) \u003d \\ DFRAC (L \\ Cdot i_ (2 ) ^ (2)) (2), \\; \\; \\; w_ (m4) \u003d \\ dfrac (l \\ cdot i_ (4) ^ (2)) (2), \\; \\; \\; w_ (m6) \u003d \\ DFRAC (L \\ Cdot i_ (6) ^ (2)) (2). \\)

The total energy of the perfect oscillatory circuit is preserved over time, since it has energy losses (no resistance). Then

\\ (W \u003d w_ (e \\, \\ max) \u003d w_ (m \\, \\ max) \u003d w_ (e2) + w_ (m2) \u003d w_ (e4) + w_ (m4) \u003d ... \\)

Thus, in perfect LC- Consture will occur periodic changes in current values i., charge q. and voltage u., Moreover, the total energy of the circuit will remain constant. In this case, they say that the contour arose free electromagnetic oscillations.

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

Thus, the occurrence of free electromagnetic oscillations in the circuit is due to the recharge of the capacitor and the emergence of self-induction EMF in the coil, which "provides" this recharge. Note that the capacitor charge q. and current power in the coil i. achieve their own maximum values Q M. and I M. at various points in time.

Free electromagnetic oscillations in the circuit occur by harmonic law:

\\ (q \u003d q_ (m) \\ cdot \\ cos \\ left (\\ \\ \\ \\ cdot t + \\ varphi _ (1) \\ right), \\; \\; \\; u \u003d u_ (m) \\ cdot \\ cos \\ left (\\ The smallest period of time during which

- Konter returns B. LCthe initial state (To the initial value of the charge of this cover), is called a period of free (own) electromagnetic oscillations in the circuit. Period of free electromagnetic oscillations in

The system is determined by the Thomson formula: LC\\ (T \u003d 2 \\ pi \\ cdot \\ sqrt (l \\ cdot c), \\; \\; \\; \\ \\ omega \u003d \\ dfrac (1) (\\ SQRT (L \\ CDOT C)). \\)

The struts of the view of the mechanical analogy, the perfect oscillatory contouration of the spring pendulum without friction, and the real - with friction. Over the priest of the friction force fluctuations in the spring pendulum fade over time.

* Conclusion of the Thomson Formula

Since the full energy is perfect

{!LANG-d5980da1a61f86f7878afd43f919f557!} LC-Conter equal to the sum of the energy of the electrostatic field of the capacitor and the magnetic field of the coil is preserved, then at any time the equality is right

\\ (W \u003d \\ dfrac (q_ (m) ^ (2)) (2c) \u003d \\ dfrac (l \\ cdot i_ (m) ^ (2)) (2) \u003d \\ DFRAC (Q ^ (2)) (2C ) + \\ DFRAC (L \\ Cdot i ^ (2)) (2) \u003d (\\ rm const). \\)

We obtain the oscillation equation in LC-The system, using the law of energy conservation. Indignantly by the expression for its total energy in time, given the fact that

\\ (W "\u003d 0, \\; \\; \\; q" \u003d i, \\; \\; \\; i "\u003d q" ", \\)

we obtain an equation describing free oscillations in the perfect circuit:

\\ (\\ left (\\ dfrac (q ^ (2)) (2c) + \\ dfrac (l \\ cdot i ^ (2)) (2) \\ right) ^ ((")) \u003d \\ DFRAC (Q) (C ) \\ cdot q "+ l \\ cdot i \\ cdot i" \u003d \\ dfrac (q) (c) \\ cdot q "+ l \\ cdot q" \\ cdot q "" \u003d 0, \\)

\\ (\\ dfrac (q) (c) + l \\ cdot q "" \u003d 0, \\; \\; \\; \\; q "" + \\ dfrac (1) (L \\ Cdot C) \\ Cdot Q \u003d 0. \\ Swiring it in the form:

\\ (q "" + \\ omega ^ (2) \\ Cdot Q \u003d 0, \\)

We notice that this is the equation of harmonic oscillations with a cyclic frequency

\\ (\\ Omega \u003d \\ DFRAC (1) (\\ SQRT (L \\ CDOT C)). \\)

Accordingly, the period of the vibrations under consideration

\\ (T \u003d \\ DFRAC (2 \\ PI) (\\ Omega) \u003d 2 \\ PI \\ CDOT \\ SQRT (L \\ CDOT C). \\)

Literature

Zhilko, V.V. Physics: studies. Manual for grade 11 general formation. shk. with rus. Yaz. Learning / V.V. Zhilko, L.G. Markovich. - Minsk: Nar. Asveta, 2009. - P. 39-43.

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