Charge in an oscillatory circuit formula. An equation describing processes in an oscillatory circuit. Period of free electrical oscillations - Knowledge Hypermarket. Topics of the Unified State Examination codifier: free electromagnetic oscillations, oscillatory circuit, forced oscillations

• Electromagnetic vibrations– these are periodic changes over time in electrical and magnetic quantities in 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- current value 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 influence of the electric field of the capacitor, an electric current will appear in the circuit, the 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 oscillatory circuit will increase from zero to some maximum value not instantly, but over a 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 creating the current continue to move by inertia. In this case, electrons arriving at the neutral plate of the capacitor give it a negative charge, electrons leaving the neutral plate give it positive charge. 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(in position 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:

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

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

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 initial state(to the initial value of the charge of a given plate) is 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.

In electrical circuits, as well as in mechanical systems such as a load on a spring or a pendulum, problems can occur. free vibrations.

Electromagnetic vibrationsare called periodic interrelated changes in charge, current and voltage.

Freeoscillations are those that occur without external influence due to the initially accumulated energy.

Forcedare called oscillations in a circuit under the influence of an external periodic electromotive force

Free electromagnetic oscillations – these are periodically repeating changes in electromagnetic quantities (q– electric charge,I– current strength,U– potential difference) occurring without energy consumption from external sources.

The simplest electrical system, capable of free vibrations, is serial RLC circuit or oscillatory circuit.

Oscillatory circuit –is a system consisting of capacitors connected in seriesC, inductorsL and a conductor with resistanceR

Consider a closed oscillatory circuit consisting of inductance L and containers WITH.

To excite oscillations in this circuit, it is necessary to impart some charge to the capacitor from the source ε . When the key K is in position 1, the capacitor is charged to voltage. After switching the key to position 2, the process of discharging the capacitor through the resistor begins R and inductor L. Under certain conditions, this process can be oscillatory in nature.

Free electromagnetic oscillations can be observed on the oscilloscope screen.

As can be seen from the oscillation graph obtained on an oscilloscope, free electromagnetic oscillations are fading, i.e. their amplitude decreases over time. This happens because part of the electrical energy at the active resistance R is converted into internal energy. conductor (the conductor heats up when electric current passes through it).

Let's consider how oscillations occur in an oscillatory circuit and what energy changes occur. Let us first consider the case when there are no losses of electromagnetic energy in the circuit ( R = 0).

If you charge the capacitor to voltage U 0, then at the initial moment of time t 1 = 0, the amplitude values ​​of voltage U 0 and charge q 0 = CU 0 will be established on the plates of the capacitor.

The total energy W of the system is equal to the energy of the electric field W el:

If the circuit is closed, current begins to flow. An emf appears in the circuit. self-induction

Due to self-induction in the coil, the capacitor is not discharged instantly, but gradually (since, according to Lenz’s rule, the resulting induced current with its magnetic field counteracts the change in the magnetic flux that caused it. That is, the magnetic field of the induced current does not allow the magnetic flux of the current to instantly increase in the circuit). In this case, the current increases gradually, reaching its maximum value I 0 at time t 2 = T/4, and the charge on the capacitor becomes zero.

As the capacitor discharges, the energy of the electric field decreases, but at the same time the energy of the magnetic field increases. The total energy of the circuit after discharging the capacitor is equal to the energy of the magnetic field W m:

At the next moment in time, the current flows in the same direction, decreasing to zero, which causes the capacitor to recharge. The current does not stop instantly after the capacitor is discharged due to self-induction (now the magnetic field of the induction current prevents the magnetic flux of the current in the circuit from instantly decreasing). At the moment of time t 3 =T/2, the charge of the capacitor is again maximum and equal to the initial charge q = q 0, the voltage is also equal to the original U = U 0, and the current in the circuit is zero I = 0.

Then the capacitor discharges again, the current flows through the inductance in the opposite direction. After a period of time T, the system returns to its initial state. The complete oscillation ends and the process repeats.

The graph of changes in charge and current strength during free electromagnetic oscillations in the circuit shows that fluctuations in current strength lag behind charge fluctuations by π/2.

At any moment of time the total energy is:

With free oscillations, periodic transformation of electrical energy occurs W e, stored in a capacitor, into magnetic energy W m coils and vice versa. If there is no energy loss in the oscillatory circuit, then the total electromagnetic energy of the system remains constant.

Free electrical vibrations are similar to mechanical vibrations. The figure shows graphs of charge changes q(t) capacitor and bias x(t) load from the equilibrium position, as well as current graphs I(t) and load speed υ( t) for one period of oscillation.

In the absence of damping, free oscillations in an electrical circuit are harmonic, that is, they occur according to the law

q(t) = q 0 cos(ω t + φ 0)

Options L And C the oscillatory circuit is determined only by the natural frequency of free oscillations and the oscillation period - Thompson’s formula

Amplitude q 0 and the initial phase φ 0 are determined initial conditions, that is, the way in which the system was brought out of equilibrium.

For fluctuations in charge, voltage and current, the following formulas are obtained:

For capacitor:

q(t) = q 0 cosω 0 t

U(t) = U 0 cosω 0 t

For inductor:

i(t) = I 0 cos(ω 0 t+ π/2)

U(t) = U 0 cos(ω 0 t + π)

Let's remember main characteristics of oscillatory motion:

q 0, U 0 , I 0 - amplitude– module highest value fluctuating magnitude

T - period– the minimum period of time after which the process is completely repeated

ν - Frequency– number of oscillations per unit time

ω - Cyclic frequency– number of oscillations in 2n seconds

φ - oscillation phase- a quantity under the cosine (sine) sign and characterizing the state of the system at any time.

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 a 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).

Topics of the Unified State Examination codifier: free electromagnetic oscillations, oscillatory circuit, forced electromagnetic oscillations, resonance, harmonic electromagnetic oscillations.

Electromagnetic vibrations- These are periodic changes in charge, current and voltage that occur in an electrical circuit. The simplest system for observing electromagnetic oscillations is an oscillatory circuit.

Oscillatory circuit

Oscillatory circuit is a closed circuit formed by a capacitor and a coil connected in series.

Let's charge the capacitor, connect the coil to it and close the circuit. Will start to happen free electromagnetic oscillations- periodic changes in the charge on the capacitor and the current in the coil. Let us remember that these oscillations are called free because they occur without any external influence - only due to the energy stored in the circuit.

The period of oscillations in the circuit will be denoted, as always, by . We will assume the coil resistance to be zero.

Let us consider in detail all the important stages of the oscillation process. For greater clarity, we will draw an analogy with the oscillations of a horizontal spring pendulum.

Starting moment: . The capacitor charge is equal to , there is no current through the coil (Fig. 1). The capacitor will now begin to discharge.

Rice. 1.

Even though the coil resistance is zero, the current will not increase instantly. As soon as the current begins to increase, a self-induction emf will arise in the coil, preventing the current from increasing.

Analogy. The pendulum is pulled to the right by an amount and released at the initial moment. The initial speed of the pendulum is zero.

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

Rice. 2.

The current increases gradually: the vortex electric field of the coil prevents the current from increasing and is directed against the current.

Analogy. The pendulum moves to the left towards the equilibrium position; the speed of the pendulum gradually increases. The deformation of the spring (aka the coordinate of the pendulum) decreases.

End of first quarter: . The capacitor is completely discharged. The current strength has reached its maximum value (Fig. 3). The capacitor will now begin recharging.

Rice. 3.

The voltage across the coil is zero, but the current will not disappear instantly. As soon as the current begins to decrease, a self-induction emf will arise in the coil, preventing the current from decreasing.

Analogy. The pendulum passes through its equilibrium position. Its speed reaches its maximum value. The spring deformation is zero.

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

Rice. 4.

The current strength decreases gradually: the eddy electric field of the coil, supporting the decreasing current, is co-directed with the current.

Analogy. The pendulum continues to move to the left - from the equilibrium position to the right extreme point. Its speed gradually decreases, the deformation of the spring increases.

End of second quarter. The capacitor is completely recharged, its charge is again equal (but the polarity is different). The current strength is zero (Fig. 5). Now the reverse recharging of the capacitor will begin.

Rice. 5.

Analogy. The pendulum has reached the far right point. The speed of the pendulum is zero. The spring deformation is maximum and equal to .

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

Rice. 6.

Analogy. The pendulum moves back: from the right extreme point to the equilibrium position.

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

Rice. 7.

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

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

Rice. 8.

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

End of the fourth quarter and the entire period: . Reverse charging of the capacitor is completed, the current is zero (Fig. 9).

Rice. 9.

This moment is identical to the moment, and this figure is identical to Figure 1. One complete oscillation took place. Now the next oscillation will begin, during which the processes will occur exactly as described above.

Analogy. The pendulum returned to its original position.

The considered electromagnetic oscillations are undamped- they will continue indefinitely. After all, we assumed that the coil resistance is zero!

In the same way, the oscillations of a spring pendulum will be undamped in the absence of friction.

In reality, the coil has some resistance. Therefore, the oscillations in a real oscillatory circuit will be damped. So, after one complete oscillation, the charge on the capacitor will be less than the original value. Over time, the oscillations will completely disappear: all the energy initially stored in the circuit will be released in the form of heat at the resistance of the coil and connecting wires.

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

Energy transformations in an oscillatory circuit

We continue to consider undamped oscillations in the circuit, considering the coil resistance to be zero. The capacitor has a capacitance and the inductance of the coil is equal to .

Since there are no heat losses, energy does not leave the circuit: it is constantly redistributed between the capacitor and the coil.

Let's take a moment in time when the charge of the capacitor is maximum and equal to , and there is no current. The energy of the magnetic field of the coil at this moment is zero. All the energy of the circuit is concentrated in the capacitor:

Now, on the contrary, let’s consider the moment when the current is maximum and equal to , and the capacitor is discharged. The energy of the capacitor is zero. All the circuit energy is stored in the coil:

At an arbitrary moment in time, when the charge of the capacitor is equal and current flows through the coil, the energy of the circuit is equal to:

Thus,

(1)

Relationship (1) is used to solve many problems.

Electromechanical analogies

In the previous leaflet about self-induction, we noted the analogy between inductance and mass. Now we can establish several more correspondences between electrodynamic and mechanical quantities.

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

(2)

Here, as you already understood, is the spring stiffness, is the mass of the pendulum, and is the current values ​​of the coordinates and speed of the pendulum, and is their greatest values.

Comparing equalities (1) and (2) with each other, we see the following correspondences:

(3)

(4)

(5)

(6)

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

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

In accordance with analogies (5) and (6), here we replace mass with inductance, and stiffness with inverse capacitance. We get:

(7)

Electromechanical analogies do not fail: formula (7) gives the correct expression for the period of oscillations in the oscillatory circuit. It's called Thomson's formula. We will present its more rigorous conclusion shortly.

Harmonic law of oscillations in a circuit

Recall that oscillations are called harmonic, if the oscillating quantity changes over time according to the law of sine or cosine. If you have forgotten these things, be sure to repeat the “Mechanical Vibrations” sheet.

The oscillations of the charge on the capacitor and the current in the circuit turn out to be harmonic. We will prove this now. But first we need to establish rules for choosing the sign for the capacitor charge and for the current strength - after all, when oscillating, these quantities will take on both positive and negative values.

First we choose positive bypass direction contour. The choice does not matter; let this be the direction counterclockwise(Fig. 10).

Rice. 10. Positive bypass direction

The current strength is considered positive class="tex" alt="(I > 0)"> , если ток течёт в положительном направлении. В противном случае сила тока будет отрицательной .!}

The charge on a capacitor is the charge on its plate to which positive current flows (i.e., the plate to which the bypass direction arrow points). IN in this case- charge left capacitor plates.

With such a choice of signs of current and charge, the following relation is valid: (with a different choice of signs it could happen). Indeed, the signs of both parts coincide: if class="tex" alt="I > 0"> , то заряд левой пластины возрастает, и потому !} class="tex" alt="\dot(q) > 0"> !}.

The quantities and change over time, but the energy of the circuit remains unchanged:

(8)

Therefore, the derivative of energy with respect to time becomes zero: . We take the time derivative of both sides of relation (8); do not forget that complex functions are differentiated on the left (If is a function of , then according to the differentiation rule complex function the derivative of the square of our function will be equal to: ):

Substituting and here, we get:

But the current strength is not a function that is identically equal to zero; That's why

Let's rewrite this as:

(9)

We have obtained a differential equation of harmonic oscillations of the form , where . This proves that the charge on the capacitor oscillates according to a harmonic law (i.e., according to the law of sine or cosine). The cyclic frequency of these oscillations is equal to:

(10)

This quantity is also called natural frequency contour; it is with this frequency that free (or, as they also say, own fluctuations). The oscillation period is equal to:

We again come to Thomson's formula.

Harmonic dependence of charge on time in general case has the form:

(11)

The cyclic frequency is found by formula (10); the amplitude and initial phase are determined from the initial conditions.

We will look at the situation discussed in detail at the beginning of this leaflet. Let the charge of the capacitor be maximum and equal (as in Fig. 1); there is no current in the circuit. Then the initial phase is , so that the charge varies according to the cosine law with amplitude:

(12)

Let's find the law of change in current strength. To do this, we differentiate relation (12) with respect to time, again not forgetting about the rule for finding the derivative of a complex function:

We see that the current strength also changes according to a harmonic law, this time according to the sine law:

(13)

The amplitude of the current is:

The presence of a “minus” in the law of current change (13) is not difficult to understand. Let's take, for example, a time interval (Fig. 2).

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

Now look at fig. 8. The current flows in the positive direction. How does our “minus” work in this case? Figure out what's going on here!

Let us depict graphs of charge and current fluctuations, i.e. graphs of functions (12) and (13). For clarity, let us present these graphs in the same coordinate axes (Fig. 11).

Rice. 11. Graphs of charge and current fluctuations

Please note: charge zeros occur at current maxima or minima; conversely, current zeros correspond to charge maxima or minima.

Using the reduction formula

Let us write the law of current change (13) in the form:

Comparing this expression with the law of charge change, we see that the current phase, equal to, is greater than the charge phase by an amount. In this case they say that the current ahead in phase charge on ; or phase shift between current and charge is equal to ; or phase difference between current and charge is equal to .

The advance of the charge current in phase is graphically manifested in the fact that the current graph is shifted left on relative to the charge graph. The current strength reaches, for example, its maximum a quarter of a period earlier than the charge reaches its maximum (and a quarter of a period exactly corresponds to the phase difference).

Forced electromagnetic oscillations

As you remember, forced oscillations arise in the system under the influence of a periodic forcing force. The frequency of forced oscillations coincides with the frequency of the driving force.

Forced electromagnetic oscillations will occur in a circuit connected to a sinusoidal voltage source (Fig. 12).

Rice. 12. Forced vibrations

If the source voltage changes according to the law:

then oscillations of charge and current occur in the circuit with a cyclic frequency (and with a period, respectively). Source AC voltage as if “imposing” its oscillation frequency on the circuit, making you forget about its own frequency.

The amplitude of forced oscillations of charge and current depends on frequency: the amplitude is greater, the closer to the natural frequency of the circuit. When resonance- a sharp increase in the amplitude of oscillations. We'll talk about resonance in more detail in the next worksheet on alternating current.

An electrical circuit consisting of an inductor and a capacitor (see figure) is called an oscillatory circuit. In this circuit, peculiar electrical oscillations can occur. Let, for example, at the initial moment of time we charge the capacitor plates with positive and negative charges, and then allow the charges to move. If the coil were missing, the capacitor would begin to discharge, an electric current would appear in the circuit for a short time, and the charges would disappear. The following happens here. First, thanks to self-induction, the coil prevents the current from increasing, and then, when the current begins to decrease, it prevents it from decreasing, i.e. supports current. As a result, the self-induction EMF charges the capacitor with reverse polarity: the plate that was initially positively charged acquires a negative charge, the second - positive. If there is no loss of electrical energy (in the case of low resistance of the circuit elements), then the value of these charges will be the same as the value of the initial charges of the capacitor 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 USE problems devoted to electromagnetic oscillations, you need to remember a number of facts and formulas regarding the oscillatory circuit. First, you need to know the formula for the period of oscillation in the circuit. Secondly, be able to apply the law of conservation of energy to an oscillatory circuit. And finally (although such tasks are rare), be able to use the dependence of the current through the coil and the voltage across the capacitor on time

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

where and is the charge on the capacitor and the current in the coil at this point in time, and is the capacitance of the capacitor and the inductance of the coil. If the electrical resistance of the circuit elements is small, then the electrical energy of the circuit (24.2) remains practically unchanged, despite the fact that the capacitor charge and the current in the coil change over time. From formula (24.4) it follows that during electrical oscillations in the circuit, energy transformations occur: at those moments in time when the current in the coil is zero, the entire energy of the circuit is reduced to the energy of the capacitor. At those moments in time when the capacitor charge is zero, the energy of the circuit is reduced to the energy of the magnetic field in the coil. Obviously, at these moments of time, the charge of the capacitor or the current in the coil reaches its maximum (amplitude) values.

During electromagnetic oscillations in the circuit, the charge of the capacitor changes over time according to the harmonic law:

standard for any harmonic vibrations. Since the current in the coil is the derivative of the capacitor charge with respect to time, from formula (24.4) we can find the dependence of the current in the coil on time

In the Unified State Examination in physics, problems on electromagnetic waves are often proposed. The minimum knowledge required to solve these problems includes an understanding of the basic properties of an electromagnetic wave and knowledge of the electromagnetic wave scale. Let us briefly formulate these facts and principles.

According to the laws of the electromagnetic field, an alternating magnetic field generates an electric field, and an alternating electric field generates a magnetic field. Therefore, if one of the fields (for example, electric) begins to change, a second field (magnetic) will arise, which then again generates the first (electric), then again the second (magnetic), etc. The process of mutual transformation of electric and magnetic fields into each other, which can propagate in space, is called an electromagnetic wave. Experience shows that the directions in which the electric and magnetic field strength vectors oscillate in an electromagnetic wave are perpendicular to the direction of its propagation. This means that electromagnetic waves are transverse. Maxwell's theory of electromagnetic field proves that an electromagnetic wave is created (emitted) electric charges when they move with acceleration. In particular, the source of the electromagnetic wave is an oscillatory circuit.

Electromagnetic wave length, its frequency (or period) and propagation speed are related by a relationship that is valid for any wave (see also formula (11.6)):

Electromagnetic waves in a vacuum propagate at speed = 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. This phenomenon is called wave dispersion. An electromagnetic wave has all the properties of waves propagating in elastic media: interference, diffraction, and Huygens’ principle is valid for it. The only thing that distinguishes an electromagnetic wave is that it does not require a medium to propagate - an electromagnetic wave can propagate in a vacuum.

In nature, electromagnetic waves are observed with frequencies that differ greatly from each other, and therefore 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 electromagnetic wave scale. Let's give brief overview this scale.

Electromagnetic waves with a frequency less than 10 5 Hz (i.e., with a wavelength greater than several kilometers) are called low-frequency electromagnetic waves. Most household electrical appliances emit waves in this range.

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

The region of the electromagnetic wave scale with frequencies lying in the range 10 12 - 4.3 10 14 Hz (and wavelengths from a few millimeters to 760 nm) is called infrared radiation (or infrared rays). The source of such radiation is the molecules of the heated substance. A person emits 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 wavelengths 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 different colors. The wave with the lowest frequency in the visible range 4.3 10 14 is perceived as red, and the highest frequency within the visible range 7.7 10 14 Hz is perceived as violet. Visible light is emitted during the transition of electrons in atoms, molecules of solids heated to 1000 °C or more.

Waves with a frequency of 7.7 10 14 - 10 17 Hz (wavelength from 390 to 1 nm) are usually called ultraviolet radiation. Ultraviolet radiation has a pronounced biological effect: it is capable of killing a number of microorganisms, can cause increased pigmentation of human skin (tanning), and with excessive irradiation in some cases it can contribute to the development of oncological diseases (skin cancer). Ultraviolet rays are contained in the radiation of the Sun and are created in laboratories with special gas-discharge (quartz) lamps.

Behind the region of ultraviolet radiation lies the region of x-rays (frequency 10 17 - 10 19 Hz, wavelength from 1 to 0.01 nm). These waves are emitted when charged particles accelerated by a voltage of 1000 V or more are decelerated in matter. They have the ability to pass through thick layers of substances that are opaque to visible light or ultraviolet radiation. Due to this property, X-rays are widely used in medicine to diagnose bone fractures and a number of diseases. X-rays have a detrimental effect on biological tissue. Thanks to this property, they can be used to treat cancer, although with excessive irradiation they are deadly to humans, causing a number of disorders in the body. Due to their very short wavelength, the wave properties of X-rays (interference and diffraction) can only be detected on structures comparable in size to atoms.

Gamma radiation (-radiation) is called electromagnetic waves with a frequency greater than 10-20 Hz (or a wavelength less than 0.01 nm). Such waves arise in nuclear processes. A special feature of -radiation is its pronounced corpuscular properties (i.e., this radiation behaves like a stream of particles). Therefore, -radiation is often spoken of as a flow of -particles.

IN problem 24.1.1 to establish correspondence between units of measurement, we use formula (24.1), from which it follows that the period of oscillation in a circuit with a capacitor of 1 F and an inductance of 1 H is equal to seconds (answer 1 ).

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

Using formula (24.1) we find the period of oscillations in the circuit given in problem 24.1.3:
(answer 4 ). Note that, according to the electromagnetic wave scale, such a circuit emits long-wave radio waves.

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

If you increase the inductance of the coil by four times ( problem 24.1.5), then according to formula (24.1) the period of oscillations in the circuit will double, and the frequency will decrease by half (answer 2 ).

According to formula (24.1), when the capacitor capacity increases fourfold ( problem 24.1.6) the period of oscillation in the circuit doubles (answer 1 ).

When the key is closed ( problem 24.1.7) in the circuit, instead of one capacitor, two identical capacitors connected in parallel will work (see figure). And since when parallel connection capacitors, their capacitances add up, then closing the switch leads to a doubling of the circuit capacitance. Therefore, from formula (24.1) we conclude that the period of oscillation increases by a factor of (answer 3 ).

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

From this formula it follows that the energy of the magnetic field in the coil oscillates with double the frequency, and, therefore, with a period half as long as the period of oscillation of charge and current (answer 1 ).

IN problem 24.1.9 We use the law of conservation of energy for the oscillatory circuit. From formula (24.2) it follows that for the amplitude values ​​of the voltage on the capacitor and the current in the coil, the relation is valid

where and are the amplitude values ​​of the capacitor charge and the current in the coil. From this formula, using relation (24.1) for the oscillation period in the circuit, we find the amplitude value of the current

answer 3 .

Radio waves are electromagnetic waves with certain frequencies. Therefore, the speed of their propagation in a vacuum is equal to the speed of propagation of any electromagnetic waves, and in particular, X-rays. This speed is the speed of light ( problem 24.2.1- answer 1 ).

As stated earlier, charged particles emit electromagnetic waves when moving with acceleration. Therefore, the wave is not emitted only with uniform and rectilinear motion ( problem 24.2.2- answer 1 ).

An electromagnetic wave is an electric and magnetic field that varies in space and time and supports each other in a special way. Therefore the correct answer is problem 24.2.3 - 2 .

From what is given in the condition tasks 24.2.4 The graph shows that the period of this wave is - = 4 µs. Therefore, from formula (24.6) we obtain m (answer 1 ).

IN problem 24.2.5 using formula (24.6) we find

(answer 4 ).

An oscillatory circuit is connected to the antenna of the electromagnetic wave receiver. The electric field of the wave acts on the free electrons in the circuit and causes them to oscillate. If the frequency of the wave coincides with the natural frequency of electromagnetic oscillations, the amplitude of oscillations in the circuit increases (resonance) and can be recorded. Therefore, to receive an electromagnetic wave, the frequency of natural oscillations in the circuit must be close to the frequency of this wave (the circuit must be tuned to the frequency of the wave). Therefore, if the circuit needs to be reconfigured from a 100 m wave to a 25 m wave ( problem 24.2.6), the natural frequency of electromagnetic oscillations in the circuit must be increased by 4 times. To do this, according to formulas (24.1), (24.4), the capacitance of the capacitor should be reduced by 16 times (answer 4 ).

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

Among those listed in problem 24.2.8 electromagnetic waves maximum frequency has x-ray radiation (answer) 2 ).

An electromagnetic wave is transverse. This means that the vectors of the electric field strength and magnetic field induction in the wave at any time are directed perpendicular to the direction of propagation of the wave. Therefore, when a wave propagates in the direction of the axis ( problem 24.2.9), the electric field strength vector is directed perpendicular to this axis. Therefore, its projection onto the axis is necessarily equal to zero = 0 (answer 3 ).

The speed of propagation of an electromagnetic wave is an individual characteristic of each medium. Therefore, when an electromagnetic wave passes from one medium to another (or from a vacuum to a medium), the speed of the electromagnetic wave changes. What can we say about the other two wave parameters included in formula (24.6) - wavelength and frequency. Will they change when a wave passes from one medium to another ( problem 24.2.10)? Obviously, the frequency of the wave does not change when moving from one medium to another. Indeed, a wave is an oscillatory process in which an alternating electromagnetic field in one medium creates and maintains a field in another medium due to these very changes. Therefore, the periods of these periodic processes (and therefore the frequencies) in one and another environment must coincide (answer 3 ). And since the speed of the wave in different media is different, it follows from the above reasoning and formula (24.6) that the wavelength changes when it passes from one medium to another.