Detailed table analogy of mechanical and electromagnetic vibrations. Analogy between mechanical and electromagnetic vibrations. Forced electromagnetic oscillations
Own undamped electromagnetic oscillations
Electromagnetic vibrations are called oscillations of electric charges, currents and physical quantities that characterize electric and magnetic fields.
Oscillations are called periodic if the values of physical quantities that change during the oscillation process are repeated at regular intervals.
The simplest type of periodic oscillations are harmonic oscillations. Harmonic oscillations are described by the equations
Or 
.
There are oscillations of charges, currents and fields that are inextricably linked with each other, and oscillations of fields that exist in isolation from charges and currents. The former take place in electrical circuits, the latter in electromagnetic waves.
Oscillatory circuit is an electrical circuit in which electromagnetic oscillations can occur.
An oscillatory circuit is any closed electrical circuit consisting of a capacitor with capacitance C, an inductor with inductance L and a resistor with resistance R, in which electromagnetic oscillations occur.
The simplest (ideal) oscillatory circuit is a capacitor and an inductor connected to each other. In such a circuit, the capacitance is concentrated only in the capacitor, the inductance is concentrated only in the coil, and, in addition, the ohmic resistance of the circuit is zero, i.e. no energy loss due to heat.
In order for electromagnetic oscillations to occur in the circuit, the circuit must be brought out of equilibrium. To do this, it is enough to charge a capacitor or excite a current in an inductor and leave it to itself.
Let us give one of the plates of the capacitor a charge + q m. Due to the phenomenon of electrostatic induction, the second plate of the capacitor will be charged with a negative charge – q m. An electric field with energy will arise in the capacitor 
.
Since the inductor is connected to a capacitor, the voltage at the ends of the coil will be equal to the voltage between the plates of the capacitor. This will lead to directional movement of free charges in the circuit. As a result, in electrical circuit contour is observed simultaneously: neutralization of charges on the capacitor plates (capacitor discharge) and orderly movement of charges in the inductor. The ordered movement of charges in the oscillatory circuit circuit is called discharge current.
Due to the phenomenon of self-induction, the discharge current will begin to increase gradually. The greater the inductance of the coil, the slower the discharge current grows.
Thus, the potential difference applied to the coil accelerates the movement of charges, and the self-induction emf, on the contrary, slows them down. Joint action potential difference And self-induction emf leads to a gradual increase discharge current . At the moment when the capacitor is completely discharged, the current in the circuit reaches maximum value I m.
This completes the first quarter of the period of the oscillatory process.
During the process of discharging a capacitor, the potential difference on its plates, the charge of the plates and the voltage electric field decrease, while the current through the inductor and induction magnetic field are increasing. The energy of the electric field of the capacitor is gradually converted into the energy of the magnetic field of the coil.
At the moment the capacitor discharges, the electric field energy will be zero, and the magnetic field energy will reach its maximum
,
where L is the inductance of the coil, I m is the maximum current in the coil.
Availability in the circuit capacitor leads to the fact that the discharge current on its plates is interrupted, the charges here are inhibited and accumulate.
On the plate towards which the current flows, positive charges, on the other plate - negative. An electrostatic field again appears in the capacitor, but now in the opposite direction. This field slows down the movement of the coil charges. Consequently, the current and its magnetic field begin to decrease. A decrease in the magnetic field is accompanied by the appearance of self-induction emf, which prevents the current from decreasing and maintains its original direction. Due to the combined action of the newly emerged potential difference and self-induction emf, the current decreases to zero gradually. The energy of the magnetic field again turns into the energy of the electric field. This completes half the period of the oscillatory process. In the third and fourth parts, the described processes are repeated as in the first and second parts of the period, but in the opposite direction. After going through all these four stages, the circuit will return to its original state. Subsequent cycles of the oscillatory process will be exactly repeated.
The following physical quantities change periodically in the oscillatory circuit:
q is the charge on the capacitor plates;
U is the potential difference across the capacitor and, therefore, at the ends of the coil;
I - discharge current in the coil;
Electric field strength;
Magnetic field induction;
W E - electric field energy;
W B - magnetic field energy.
Let's find the dependences of q, I, , W E, W B on time t.
To find the law of charge change q = q(t), it is necessary to compose for it differential equation and find a solution to this equation.
Since the circuit is ideal (i.e. it does not emit electromagnetic waves and does not generate heat), its energy, consisting of the sum of the magnetic field energy W B and the electric field energy W E , remains unchanged at any time.
where I(t) and q(t) are the instantaneous values of current and charge on the capacitor plates.
Having designated 
, we obtain a differential equation for the charge
The solution to the equation describes the change in charge on the capacitor plates over time.
,
where is the amplitude value of the charge; - initial phase; - cyclic oscillation frequency, 
- oscillation phase.
Oscillations of any physical quantity described by an equation are called undamped natural oscillations. The quantity is called the natural cyclic frequency of oscillations. The oscillation period T is the shortest period of time after which physical quantity takes on the same value and has the same speed.
The period and frequency of natural oscillations of the circuit are calculated using the formulas:
Expression 
called Thomson's formula.
Changes in potential difference (voltage) between capacitor plates over time
, Where 
- voltage amplitude.
The dependence of the current strength on time is determined by the relation –
Where 
- current amplitude.
The dependence of the self-induction emf on time is determined by the relation –
Where 
- amplitude of self-induction emf.
The dependence of the electric field energy on time is determined by the relation
Where 
- amplitude of the electric field energy.
The time dependence of the magnetic field energy is determined by the relation
Where 
- amplitude of magnetic field energy.
The expressions for the amplitudes of all changing quantities include the charge amplitude q m. This value, as well as the initial phase of oscillations φ 0, are determined by the initial conditions - the charge of the capacitor and the current in 
circuit at the initial time t = 0.
Dependencies 
from time t are shown in Fig.
In this case, the oscillations of the charge and the potential difference occur in the same phases, the current lags in phase from the potential difference by , the frequency of oscillations of the energies of the electric and magnetic fields is twice the frequency of oscillations of all other quantities.
Electrical and magnetic phenomena are inextricably linked. Change electrical characteristics any phenomenon entails a change in its magnetic characteristics. Electromagnetic vibrations are of particular practical value.
Electromagnetic vibrations– these are interrelated changes in the electric and magnetic fields, at which the values of the quantities characterizing the system ( electric charge, current, voltage, energy), are repeated to varying degrees.
It should be noted that between fluctuations of different physical nature there is an analogy. They are described by the same differential equations and functions. Therefore, the information obtained in the study of mechanical vibrations turns out to be useful in the study of electromagnetic vibrations.
In modern technology, electromagnetic oscillations and waves play a greater role than mechanical ones, as they are used in communication devices, television, radar, and in various technological processes that have determined scientific and technological progress.
Electromagnetic oscillations are excited in oscillatory system, called oscillatory circuit. It is known that any conductor has electrical resistance R, electrical capacity WITH and inductance L, and these parameters are distributed along the length of the conductor. Focused parameters R, WITH, L possess a resistor, capacitor and coil respectively.
An oscillatory circuit is a closed electrical circuit consisting of a resistor, capacitor and coil (Fig. 4.1). This system is similar to a mechanical pendulum.
The circuit is in a state of equilibrium if there are no charges or currents in it. To unbalance the circuit, it is necessary to charge the capacitor (or excite induced current using a changing magnetic field). Then an electric field with intensity will arise in the capacitor. When the key is closed TO current will flow in the circuit, as a result the capacitor will discharge, the energy of the electric field will decrease, and the energy of the magnetic field of the inductor will increase.
Rice. 4.1 Oscillatory circuit
At some point in time, equal to a quarter of the period, the capacitor will be completely discharged, and the magnetic field will reach its maximum. This means that the energy of the electric field has been converted into energy of the magnetic field. Since the currents supporting the magnetic field have disappeared, it will begin to decrease. The decreasing magnetic field causes a self-induction current, which, according to Lenz's law, is directed in the same way as the discharge current. Therefore, the capacitor will be recharged and an electric field will appear between its plates with a intensity opposite to the original one. After a time equal to half the period, the magnetic field will disappear, and the electric field will reach its maximum.
Then all processes will occur in the opposite direction and after a time equal to the oscillation period, the oscillatory circuit will return to its original state with a capacitor charge. Consequently, electrical oscillations occur in the circuit.
For a complete mathematical description of the processes in the circuit, it is necessary to find the law of change in one of the quantities (for example, charge) over time, which, when using the laws of electromagnetism, will allow one to find the patterns of change in all other quantities. Functions that describe changes in quantities characterizing processes in the circuit are the solution to a differential equation. To compile it, Ohm's law and Kirchhoff's rules are used. However, they are performed for direct current.
An analysis of the processes occurring in an oscillatory circuit has shown that the laws of direct current can also be applied to a time-varying current that satisfies the quasi-stationary condition. This condition is that during the propagation of the disturbance to the most remote point of the circuit, the current strength and voltage change insignificantly, then the instantaneous values electrical quantities at all points of the chain are almost identical. Since the electromagnetic field propagates in a conductor at the speed of light in a vacuum, the propagation time of disturbances is always less than the period of current and voltage oscillations.
In the absence of an external source in the oscillatory circuit, free electromagnetic vibrations.
According to Kirchhoff's second rule, the sum of the voltages across the resistor and the capacitor is equal to electromotive force, V in this case EMF of self-induction that occurs in a coil when a changing current flows through it
Taking into account that , and, therefore, , we present expression (4.1) in the form:
. (4.2)
Let us introduce the following notations: , .
Then equation (4.2) will take the form:
. (4.3)
The resulting expression is a differential equation that describes the processes in the oscillatory circuit.
In the ideal case, when the resistance of the resistor can be neglected, free vibrations in the circuit are harmonic.
In this case, the differential equation (4.3) will take the form:
and its solution will be a harmonic function
, (4.5)
>> Analogy between mechanical and electromagnetic vibrations
§ 29 ANALOGY BETWEEN MECHANICAL AND ELECTROMAGNETIC VIBRATIONS
Electromagnetic oscillations in the circuit are similar to free mechanical oscillations, for example, with oscillations of a body mounted on a spring (spring pendulum). The similarity does not relate to the nature of the quantities themselves, which change periodically, but to the processes of periodic change in various quantities.
During mechanical vibrations, the coordinates of the body periodically change X and the projection of its speed x, and with electromagnetic oscillations the charge q of the capacitor and the current strength change i in the chain. The identical nature of the change in quantities (mechanical and electrical) is explained by the fact that there is an analogy in the conditions under which mechanical and electromagnetic vibrations arise.
The return to the equilibrium position of the body on the spring is caused by the elastic force F x extr, proportional to the displacement of the body from the equilibrium position. The proportionality coefficient is the spring stiffness k.
The discharge of the capacitor (the appearance of current) is caused by the voltage between the plates of the capacitor, which is proportional to the charge q. The proportionality coefficient is the reciprocal of the capacitance, since u = q.
Just as, due to inertia, a body only gradually increases speed under the influence of a force and this speed does not immediately become zero after the force ceases, the electric current in a coil due to the phenomenon of self-induction increases gradually under the influence of voltage and does not disappear immediately when this voltage becomes equal to zero. The inductance of the circuit L plays the same role as the body mass m during mechanical vibrations. Respectively kinetic energy body is similar to the energy of the magnetic field of the current
Charging a capacitor from a battery is similar to imparting potential energy to a body attached to a spring when the body is displaced a distance x m from the equilibrium position (Fig. 4.5, a). Comparing this expression with the energy of the capacitor, we note that the stiffness k of the spring plays the same role during mechanical vibrations as the reciprocal of capacitance during electromagnetic vibrations. In this case, the initial coordinate x m corresponds to the charge q m.
The appearance of current i in the electrical circuit corresponds to the appearance of body speed x in the mechanical oscillatory system under the action of the elastic force of the spring (Fig. 4.5, b).
The moment in time when the capacitor is discharged and the current reaches its maximum is similar to the moment in time when the body passes through the equilibrium position at maximum speed (Fig. 4.5, c).
Next, the capacitor will begin to recharge during electromagnetic oscillations, and the body, during mechanical oscillations, will begin to shift to the left of the equilibrium position (Fig. 4.5, d). After half the period T has passed, the capacitor will be completely recharged and the current will become zero.
With mechanical vibrations, this corresponds to the deflection of the body to the extreme left position, when its speed is zero (Fig. 4.5, e).
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Oscillation of a load on a spring
During mechanical vibrations of a body on a spring, the coordinate of the body will periodically change. In this case, the projection of the body’s velocity onto the Ox axis will change. In electromagnetic oscillations over time periodic law the charge q of the capacitor will change, and the current in the oscillatory circuit circuit will change.
The quantities will have the same pattern of change. This happens because there is an analogy between the conditions in which oscillations occur. When we remove the load on the spring from the equilibrium position, an elastic force F ex. arises in the spring, which tends to return the load back to the equilibrium position. The proportionality coefficient of this force will be the spring stiffness k.
When the capacitor discharges, a current appears in the oscillatory circuit circuit. Discharge is due to the fact that there is a voltage u across the capacitor plates. This voltage will be proportional to the charge q of any of the plates. The proportionality coefficient will be the value 1/C, Where C is the capacitance of the capacitor.
When a load moves on a spring, when we release it, the speed of the body increases gradually, due to inertia. And after the cessation of force, the speed of the body does not immediately become zero, it also gradually decreases.
Oscillatory circuit
The same is true in an oscillatory circuit. Electricity in a coil under the influence of voltage does not increase immediately, but gradually, due to the phenomenon of self-induction. And when the voltage stops acting, the current does not immediately become zero.
That is, in an oscillatory circuit, the inductance of the coil L will be similar to the body mass m, when the load on the spring oscillates. Consequently, the kinetic energy of the body (m*V^2)/2 will be similar to the energy of the magnetic field of the current (L*i^2)/2.
When we remove the load from the equilibrium position, we impart to the mind some potential energy (k*(Xm)^2)/2, where Xm is the displacement from the equilibrium position.
In an oscillatory circuit, the role of potential energy is played by the charge energy of the capacitor q^2/(2*C). We can conclude that the spring stiffness in mechanical vibrations will be similar to the value 1/C, where C is the capacitance of the capacitor in electromagnetic vibrations. And the coordinate of the body will be similar to the charge of the capacitor.
Let's take a closer look at the oscillation processes in the following figure.
picture
(a) We impart potential energy to the body. By analogy, we charge a capacitor.
(b) Let go of the ball potential energy begins to decrease, the speed of the ball increases. By analogy, the charge on the capacitor plate begins to decrease, and current strength appears in the circuit.
(c) Equilibrium position. There is no potential energy, the body speed is maximum. The capacitor is discharged, the current in the circuit is maximum.
(e) The body deviated to its extreme position, its speed became equal to zero, and the potential energy reached its maximum. The capacitor charged again, the current in the circuit became zero.
Themes Unified State Exam 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 An oscillatory circuit is used to observe electromagnetic oscillations.
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 oscillations of the 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 this moment 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: . The reverse recharging 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 is 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 counterclock-wise(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 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 got the differential equation harmonic vibrations kind , 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 period of oscillation is:
We again come to Thomson's formula.
The harmonic dependence of charge on time in the 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). The AC voltage source seems to “impose” 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.