To excite the oscillation circuit, the capacitor is pre-charged, imparting a charge to its plates ±q. Then at the initial moment of time t= 0 (Fig. 19, A) An electric field will arise between the plates of the capacitor. If you close a capacitor to an inductor, the capacitor will begin to discharge, and a current will flow in the circuit, increasing over time. I. When the capacitor is completely discharged, the energy of the electric field of the capacitor will be completely converted into the energy of the magnetic field of the coil (Fig. 19, b). Starting from this moment, the current in the circuit will decrease, and, consequently, the magnetic field of the coil will begin to weaken, then, according to Faraday’s law, a current is induced in it, which flows in accordance with Lenz’s rule in the same direction as the discharge current of the capacitor. The capacitor will begin to recharge, an electric field will arise, tending to weaken the current, which will eventually go to zero, and the charge on the capacitor plates will reach a maximum (Fig. 19, V). Next, the same processes will begin to occur in the opposite direction (Fig. 19, G), and the system at the time t=T (T– period of oscillation) will return to its original state (Fig. 19, A). After this, the repetition of the considered cycle of discharging and charging the capacitor will begin, that is, periodic undamped oscillations of the charge amount will begin q on the capacitor plates, voltage U C on the capacitor and current I, flowing through the inductor. According to Faraday's law, voltage U C on the capacitor is determined by the rate of change of current in the inductor of an ideal circuit, that is:
Based on the fact that U C =q/C, A I=dq/dt, we get differential equation of free undamped harmonic oscillations charge magnitude q on the capacitor plates:
or 
.
The solution to this differential equation is the function q(t), that is equation of free undamped harmonic oscillations charge magnitude q on the capacitor plates:
Where q(tt;
q 0 – amplitude of charge oscillations on the capacitor plates;
– circular (or cyclic) oscillation frequency () ;
2 /T(T– period of oscillation, 
–Thomson's formula);
– phase of oscillations at the moment of time t;
– initial phase of oscillations, that is, the phase of oscillations at the moment of time t=0.
Equation of free damped harmonic oscillations. In a real oscillatory circuit, it is taken into account that in addition to the inductance coil L, capacitor with a capacity WITH, the circuit also contains a resistor with a resistance R,different from zero, which is the reason for the damping of oscillations in a real oscillatory circuit. Available damped oscillations– oscillations, the amplitude of which decreases over time due to energy losses by the real oscillatory system.
For a circuit of a real oscillatory voltage circuit on a series-connected capacitor with a capacitance WITH and a resistor with resistance R fold up. Then, taking into account Faraday’s law for the circuit of a real oscillatory circuit, we can write:
,
where is the electromotive force of self-induction in the coil;
U C– voltage across the capacitor ( U C =q/C);
IR– voltage across the resistor.
Based on the fact that I=dq/dt, we get differential equation of free damped harmonic oscillations charge magnitude q on the capacitor plates:
or 
,
where is the vibration damping coefficient () , .
q(t), that is equation of free damped harmonic oscillations charge magnitude q on the capacitor plates:
Where q(t) – the amount of charge on the capacitor plates at the moment of time t;
– amplitude of damped oscillations of the charge at the moment of time t;
q 0 – initial amplitude of damped charge oscillations;
– circular (or cyclic) oscillation frequency ( 
);
– phase of damped oscillations at the moment of time t;
– the initial phase of damped oscillations.
Period of free damped oscillations in a real oscillatory circuit:
.
Forced electromagnetic oscillations. In order to obtain undamped oscillations in a real oscillatory system, it is necessary to compensate for energy losses during the oscillation process. Such compensation in a real oscillatory circuit is possible with the help of an external alternating voltage periodically varying according to a harmonic law U(t):
.
In this case differential equation of forced electromagnetic oscillations will take the form:
or 
.
The solution to the resulting differential equation is the function q(t):
In steady state, forced oscillations occur with a frequency w and are harmonic, and the amplitude and phase of the oscillations are determined by the following expressions:
; 
.
It follows that the amplitude of oscillations of the charge value has a maximum at the resonant frequency of the external source:
.
The phenomenon of a sharp increase in the amplitude of forced oscillations as the frequency of the forcing alternating voltage approaches a frequency close to the frequency is called resonance.
Topic 10. Electromagnetic waves
According to Maxwell's theory, electromagnetic fields can exist in the form of electromagnetic waves, the phase velocity the distribution of which is determined by the expression:
,
where and are the electric and magnetic constants, respectively,
e And m– electric and magnetic permeability of the medium, respectively,
With– speed of light in vacuum () .
In a vacuum ( e= 1, m= l) the speed of propagation of electromagnetic waves coincides with the speed of light( With), which is consistent with Maxwell's theory that
that light is electromagnetic waves.
According to Maxwell's theory electromagnetic waves are transverse, that is, the vectors and strengths of the electric and magnetic fields are mutually perpendicular and lie in a plane perpendicular to the vector
wave propagation speed, and the vectors , and form a right-handed screw system (Fig. 20).
From Maxwell’s theory it also follows that in an electromagnetic wave the vectors and oscillate in the same phases (Fig. 20), that is, the strength values E And N electric and magnetic fields simultaneously reach a maximum and simultaneously turn to zero, and the instantaneous values E And N are related by the relation: 
.
Equation of a plane monochromatic electromagnetic wave(indexes at And z at E And N They only emphasize that the vectors and are directed along mutually perpendicular axes in accordance with Fig. 20).
Oscillations movements or processes that are characterized by a certain repeatability over time are called. Oscillatory processes are widespread in nature and technology, for example, the swinging of a clock pendulum, alternating electric current, etc. When the pendulum oscillates, the coordinate of its center of mass changes; in the case of alternating current, the voltage and current in the circuit fluctuate. The physical nature of vibrations can be different, therefore, there are mechanical, electromagnetic, etc. vibrations. However, different oscillatory processes are described by the same characteristics and the same equations. Hence the expediency common approach to the study of vibrations of different physical nature.
Oscillations are called free, if they occur only under the influence of internal forces acting between the elements of the system, after the system is taken out of equilibrium by external forces and left to itself. Free vibrations always damped oscillations , because in real systems energy losses are inevitable. In the idealized case of a system without energy loss, free oscillations (continuing as long as desired) are called own.
The simplest type of free undamped oscillations are harmonic vibrations - oscillations in which the oscillating quantity changes over time according to the law of sine (cosine). Vibrations found in nature and technology often have a character close to harmonic.
Harmonic oscillations are described by an equation called the harmonic oscillation equation:
Where A- amplitude of oscillations, maximum value of the oscillating quantity X; - circular (cyclic) frequency of natural oscillations; - initial phase of oscillation at the moment of time t= 0; - phase of oscillation at the moment of time t. The oscillation phase determines the value of the oscillating quantity at a given time. Since the cosine varies from +1 to -1, then X can take values from + A before - A.
Time T during which the system completes one complete oscillation is called period of oscillation. During T the oscillation phase is incremented by 2 π , i.e.
Where . (14.2)
The reciprocal of the oscillation period
i.e., the number of complete oscillations performed per unit time is called the oscillation frequency. Comparing (14.2) and (14.3) we get
The unit of frequency is hertz (Hz): 1 Hz is the frequency at which one complete oscillation occurs in 1 s.
Systems in which free vibrations can occur are called oscillators . What properties must a system have in order for free vibrations to occur in it? The mechanical system must have stable equilibrium position, upon exiting which appears restoring force directed towards the equilibrium position. This position corresponds, as is known, to the minimum potential energy of the system. Let us consider several oscillatory systems that satisfy the listed properties.
Changes in any quantity are described using the laws of sine or cosine, then such oscillations are called harmonic. Let's consider a circuit consisting of a capacitor (which was charged before being included in the circuit) and an inductor (Fig. 1).
Picture 1.
The harmonic vibration equation can be written as follows:
$q=q_0cos((\omega )_0t+(\alpha )_0)$ (1)
where $t$ is time; $q$ charge, $q_0$-- maximum deviation of charge from its average (zero) value during changes; $(\omega )_0t+(\alpha )_0$- oscillation phase; $(\alpha )_0$- initial phase; $(\omega )_0$ - cyclic frequency. During the period, the phase changes by $2\pi $.
Equation of the form:
equation of harmonic oscillations in differential form for an oscillatory circuit that will not contain active resistance.
Any type of periodic oscillations can be accurately represented as a sum of harmonic oscillations, the so-called harmonic series.
For the oscillation period of a circuit that consists of a coil and a capacitor, we obtain Thomson’s formula:
If we differentiate expression (1) with respect to time, we can obtain the formula for the function $I(t)$:
The voltage across the capacitor can be found as:
From formulas (5) and (6) it follows that the current strength is ahead of the voltage on the capacitor by $\frac(\pi )(2).$
Harmonic oscillations can be represented both in the form of equations, functions and vector diagrams.
Equation (1) represents free undamped oscillations.
Damped Oscillation Equation
The change in charge ($q$) on the capacitor plates in the circuit, taking into account the resistance (Fig. 2), will be described by a differential equation of the form:
Figure 2.
If the resistance that is part of the circuit $R\
where $\omega =\sqrt(\frac(1)(LC)-\frac(R^2)(4L^2))$ is the cyclic oscillation frequency. $\beta =\frac(R)(2L)-$damping coefficient. The amplitude of damped oscillations is expressed as:
If at $t=0$ the charge on the capacitor is equal to $q=q_0$ and there is no current in the circuit, then for $A_0$ we can write:
The phase of oscillations at the initial moment of time ($(\alpha )_0$) is equal to:
When $R >2\sqrt(\frac(L)(C))$ the change in charge is not an oscillation, the discharge of the capacitor is called aperiodic.
Example 1
Exercise: The maximum charge value is $q_0=10\ C$. It varies harmonically with a period of $T= 5 s$. Determine the maximum possible current.
Solution:
As a basis for solving the problem we use:
To find the current strength, expression (1.1) must be differentiated with respect to time:
where the maximum (amplitude value) of the current strength is the expression:
From the conditions of the problem we know the amplitude value of the charge ($q_0=10\ C$). You should find the natural frequency of oscillations. Let's express it as:
\[(\omega )_0=\frac(2\pi )(T)\left(1.4\right).\]
In this case, the desired value will be found using equations (1.3) and (1.2) as:
Since all quantities in the problem conditions are presented in the SI system, we will carry out the calculations:
Answer:$I_0=12.56\ A.$
Example 2
Exercise: What is the period of oscillation in a circuit that contains an inductor $L=1$H and a capacitor, if the current strength in the circuit changes according to the law: $I\left(t\right)=-0.1sin20\pi t\ \left(A \right)?$ What is the capacitance of the capacitor?
Solution:
From the equation of current fluctuations, which is given in the conditions of the problem:
we see that $(\omega )_0=20\pi $, therefore, we can calculate the Oscillation period using the formula:
\ \
According to Thomson's formula for a circuit that contains an inductor and a capacitor, we have:
Let's calculate the capacity:
Answer:$T=0.1$ c, $C=2.5\cdot (10)^(-4)F.$
