SHARE WITH FRIENDS:
Electromagnetic vibrations and waves
In the study of vibrations, we said that depending on their physical nature, vibrations are divided into two, i.e., mechanical and electromagnetic vibrations.
Electromagnetic oscillations refer to interrelated periodic changes of charges, currents, electric and magnetic field strengths.
Similar processes occur when electrical oscillations are generated in a system called an oscillation circuit.
|
|
|
|
The oscillation circuit is an integral part of any radio equipment. In radio conductors, the oscillation circuit serves to radiate electromagnetic waves in space, and in radio receivers (radio receivers) to isolate the necessary part of the spectrum of electromagnetic waves.
A capacitor and an inductance C are connected by conductors to each other as an oscillation circuit L is called an electrical circuit consisting of (Fig. 1).
In the ideal vibration contour (active resistance R equal to zero) we will consider the formation of oscillations. To create an oscillation in such a circuit, it is necessary to give a certain amount of electric charge to the capacitor plates or to induce an electric current in the inductance coil.
Suppose we opened the circuit and charged the capacitor (Fig. 2a). An electronic field is formed between the capacitor plates, the energy of which is equal to:
(1)
|
|||||||||||
|
|
|
|
|
|
|
|
|
|
|
L L L
a a a
-
a) b) c) g) d)
Figure 2.
bunda C – capacitor capacity; U0 – the maximum tension between the coatings.
This state of the oscillation contour is similar to the state of a mathematical pendulum deviated from the equilibrium state by a small angle α.
We connect the capacitor C to the inductance L (Fig. 2b). The capacitor begins to discharge and its electric field decreases. At the same time, an electric current appears in the circuit, and as a result, a magnetic field is formed in the inductive coil.
In an ideal circuit, after a quarter period, the electric field energy is converted into full magnetic field energy:
(2)
bunda L – coil inductance; J – the maximum value of the current flowing through the coil. In this case, the voltage between capacitor plates is zero, U=0. This state of the oscillation contour corresponds to the state of the mathematical pendulum during transition from the equilibrium state. In this case, the potential energy of the system turns into full kinetic energy.
Then the magnetic field must quickly decrease to zero because there is no current to support it. The changing magnetic field creates an induction current that, according to Lenz's law, charges the capacitor's decreasing discharge current. Thus, the current flows in this direction and recharges the capacitor. As soon as the capacitor recharges, the current in the circuit is exhausted. Therefore, after a time equal to half a period, the magnetic field disappears, that is, the energy of the magnetic field is completely converted into the energy of the electric field (Fig. 2v). This position of the oscillation contour is similar to the position of a mathematical pendulum tilted in the opposite direction by an angle α.
After that, the capacitor begins to charge again, the current begins to flow in the circuit again, but the direction of this current is opposite to the previous one, after some time the capacitor is completely discharged, the electric field energy turns into magnetic field energy (Fig. 2g), t From time =T, the state of the contour (Fig. 2d) will be the same as the initial state. After that, the whole process is repeated.
Oscillations occur in the circuit, in which there are periodic changes in the voltage and current between the capacitor plates. In this way, the energy of the electric field turns into the energy of the magnetic field, and vice versa, the energy of the magnetic field turns into the energy of the electric field, that is, electromagnetic vibrations occur. If the resistance of the circuit is zero, the process of converting electric field energy into magnetic field energy and its opposite can continue indefinitely, that is, inexhaustible electromagnetic oscillations occur. These oscillations are said to be spontaneous or free oscillations because they occur without external forcing forces.
Using the analogy between mechanical and electrical oscillations, one can find the frequency of specific oscillations in the circuit. Looking at the oscillation of a spring pendulum, we found that its oscillation period depends on the mass of the load and the stiffness of the spring. In the oscillation circuit, the inductance L plays the role of mass, and the inverse of the capacitance 1/S plays the role of unity.
Thus, the period of free undamped electromagnetic oscillations in the oscillation circuit is determined from the Thomson formula:
(3)
Eigenfrequency and eigenperiodic frequency of electromagnetic oscillations, knowing the period of oscillation ω0 can be defined as:
(4)
(5)
The alternating electric and magnetic fields generated in the oscillation contour are located in the space where the contour is located. Such a contour is called a closed oscillation contour.
All real circuits have non-zero resistance R. Therefore, free electromagnetic oscillations in the circuit are damped. We see an electric circuit consisting of a series-connected capacitor S, a coil with an inductance L, an electrical resistance R, and a switch K (Fig. 3).
|
|
|||||
|
L
|
|
Figure 3.
If we charge the capacitor up to the potential difference without the switch connected, and then connect the switch, the capacitor will begin to discharge. As a result, a time-varying current J begins to flow through the circuit. For the circuit shown in Fig. 3, we determine the relation of current strength to time t. For simplicity, we assume that the electrical resistance of the coil, wires, and switch is zero. Based on Ohm's law, we write the following for the 1L R 2 part of the chain:
(6)
bunda J, Dph, ε – respectively, the instantaneous value of the current in the circuit, the potential difference between the 1 and 2 covers of the capacitor, and the algebraic sum of the EYUKs placed in the considered part of the circuit. In the part of the circuit 1L R 2, only self-induction EYUK is formed when an alternating current passes through the coil.
that is why
(7)
then equation (6) takes the following form:
(8)
If q is the charge on the first layer of the capacitor, then the current in the circuit is as follows.
(9)
The reason for the minus sign in the formula (9) is that the direction of the current taken as positive shown in Figure 3 when constructing the equation (b) corresponds to the reduction of the positive charge on the first cover of the capacitor ().
The potential difference between capacitor plates is equal to:
(10)
Putting expressions (9) and (10) into equation (8), we get:
(11)
This differential equation is similar to the differential equation for damped oscillations of a load suspended on a spring with the form:
(12)
Instead of the mass of the load, the inductance L of the circuit, instead of the coefficient of resistance, the resistance of the circuit R, instead of the elastic coefficient of the spring, there is the inverse of its capacity - 1/S.
As we know in the department of mechanics, the solution of equation (12) has the following form:
(13)
(14)
ω – cyclic frequency of damping oscillations of the load in the spring;
A0 va φ0 – initial value of amplitude and phase.
In formulas (13) and (14), we find the solution of differential equation (1) by replacing m, r and k by L, R and 11/S.
(15)
(16)
Thus, when a charged capacitor is connected to a circuit consisting of an inductance and an electrical resistance connected in series, the charge on the capacitor produces damping oscillations. That's why the chain being built is called a vibration circuit.
(17)
The quantity b is called extinction coefficient. It can be seen from (15) that the amplitude of oscillations of the capacitor charge q, A, is equal to:
(18)
The potential difference between the plates of the capacitor is proportional to the charge q. that is why
(19)
(15) and from the formulas we derive the following expression for the current strength in the oscillation circuit:
(20)
At the beginning of time (t=0) the charge of the capacitor is q=q0 we assume that At this time, there is no current in the chain, and from formulas (15) and (20) we get:
va
In this case, the initial phase is a0 and initial amplitude A0 we form the following relation for s:
(21)
(22)
Thus, the initial phase and amplitude of oscillations in the circuit depend on its parameters: capacitance, inductance and resistance.
The period T of undisturbed oscillations in the contour is equal to:
(23)
A changing electric current in the loop creates a changing magnetic field. At the same time, the electric field of the capacitor also changes. Therefore, the free oscillations of the capacitor charge and the current in the circuit are called free electromagnetic oscillations. The energy of these vibrations is equal to the electric energy of the charged capacitor in the initial state. Then, the electromagnetic oscillations in the circuit gradually decrease as the Joule-Lens heat is released as the current flows. After that, the energy of electromagnetic vibrations dissipates and fades. In order to generate continuous electromagnetic oscillations, energy must be supplied to the circuit from the outside to replenish the energy lost due to Joule-Lens heat. In this case, we no longer deal with free, but with forced electromagnetic oscillations. To generate such oscillations, it is necessary to connect a current source with a periodically changing EYUK to the oscillation circuit (Fig. 4).
(24)
|
|
|
|
Figure 4.
In this case, forced oscillations are formed in the circuit, the frequency of which is determined by the frequency of the current source EYUK ω. Amplitude of the current in the circuit depends not only on the parameters of the circuit, that is, R, L, C and the frequency of ЕУК. If ω is the frequency of specific oscillations of the vibration contour ω0 is equal to or close to, the phenomenon of a sharp increase in the amplitude of the current in the circuit occurs, that is, a resonance phenomenon occurs. The resonant frequency for current is:
(25)
The resonance frequency does not depend on the active resistance of the circuit.
Currently, self-oscillating systems are used to create continuous oscillations.
Alternating electric and magnetic fields are related to each other. They apply to each other and propagate in space as an electromagnetic wave, independent of the source that generated them.
The electric field strength Ye and the magnetic field induction V lead to the space propagation of a periodically changing variable electromagnetic field, called an electromagnetic wave. The graph of an electromagnetic wave can be represented as sinusoids lying in two mutually perpendicular planes. One sinusoid reflects the vibration of the electric field strength vector Ye, and the second one reflects the vibration of the magnetic induction vector V (Fig. 5).
z
|
|
y
|
|
x
Figure 5.
Electric and magnetic field lines of force are mutually perpendicular, so vectors E and B lie in a plane perpendicular to each other and are perpendicular to the direction of their propagation.
Thus, electromagnetic waves are transverse waves.
According to Maxwell's theory, the propagation speed of electromagnetic waves is a finite quantity, which is determined by the electric and magnetic properties of the medium in which the wave propagates:
(26)
Bunda ε0 va m0 - electric and magnetic constants;
ε va m — relative dielectric and magnetic absorbencies of the medium.
If an electromagnetic wave is propagating in a vacuum, ε= 1, m= 1, so the speed of propagation of an electromagnetic wave in space:
m / s
The speed of propagation of an electromagnetic wave in space is equal to the speed of propagation of light in space:
m / s
If the propagation speed of an electromagnetic wave in a homogeneous medium v If we say that the oscillation period is T and the wavelength is λ,
(27)
will be For space
(28)
The speed of the wave is medium ε va m since it depends on , when moving from one environment to another v and λ changes, but the frequency remains the same.
An electromagnetic wave propagating in space carries energy W. Electromagnetic field energy means the sum of the energies of electric and magnetic fields:
(29)
In this case, the energy density of the electromagnetic field is the sum of the energy densities of the electric and magnetic fields:
(30)
In order for an electromagnetic wave to propagate with the speed of light C, the energy flow of the following amount passes through a unit surface per unit time:
(31)
The following follows from Maxwell's theory
(32)
using the relation (31), the formula can be reduced to the following form:
(33)
The vector S, whose direction is the same as the direction of electromagnetic wave propagation and defined by the formula (33), is called the Umov-Poynting vector. It is numerically equal to the energy carried by an electromagnetic wave over a unit surface per unit time.
The researches of scientists such as PNLebedev, AAGglagoleva-Arkadeva show that all the properties of electromagnetic waves are the same as the properties of light. From this comes such an important conclusion that light consists of an electromagnetic wave. Further research shows that not only visible light, but also infrared and ultraviolet radiation, X-rays and gamma rays have the nature of electromagnetic waves. So the frequency and wavelengths of electromagnetic waves occupy a very wide range.
All types of electromagnetic waves propagate in space with the same speed. They differ only in wavelengths:
Bunda C – speed of light; — frequency.
Radio waves and ultrashort waves (UTQ) have wavelengths from several kilometers to several centimeters. They are created using vibrators of various constructions. Infrared radiation, visible light and ultraviolet rays are emitted by objects heated to different temperatures. The higher the temperature, the shorter the wavelength of the electromagnetic waves they emit. X-rays are produced by the sudden braking of charged particle electrons. Gamma rays are emitted as a result of radioactive decay of atomic nuclei.
The idea of using electromagnetic waves to transmit signals over long distances was first proposed by ASPopov in 1889.
Radio communication is the transmission of information over a distance using electromagnetic waves. Manifestations of radio communication are radio broadcasting (transmission of words and music) and television broadcasting (transmission of images).
A functional diagram of a modern radio transmitter and radio receiver is shown in Figure 6.
4 5
|
|
|||||
|
||||
|
6
An inexhaustible (1) vibration generator produces high-frequency vibrations. Sound vibrations are converted into electrical vibrations using a microphone. (1) Vibrations and sound vibrations from the generator fall into a device (2) called a modulator. In this device, the amplitude (amplitude modulation) or frequency (frequency modulation) of the vibrations produced by the generator changes under the influence of sound vibrations. An example of amplitude modulation is shown in Figure 7.
v
t a)
v
b)
v
c)
Figure 7.
Figure 7a shows the generator signal, Figure 7b shows the signal from the microphone in the modulator, and Figure 7v shows the modulated signal. Modulation for speech and music transmission is at the audio frequency (10¸13)×103 Gs is implemented.
After being amplified in the amplifier (3), the modulated vibrations are transmitted to the transmitting antenna (4). This antenna is an open oscillating circuit that emits electromagnetic waves in the air.
A radio receiver is located at a certain distance from the radio transmitter. Electromagnetic waves arrive at the antenna (5) of the radio receiver and generate electromagnetic vibrations in the circuit (5b). (5b) a capacitor whose capacity changes is connected in the circuit. By changing the capacitance of the capacitor, the natural frequency of the circuit can be changed. In this way, the receiver circuit is brought into resonance with the frequency of the received electromagnetic waves. Received high-frequency vibrations (7) go to the amplifier and from there to the detector. In the detector, the process of converting high-frequency modulated vibrations into low-frequency vibrations occurs. Then the low-frequency vibrations (9) are amplified using an amplifier and transmitted to the speaker. The information coming to the microphone is reproduced using the speaker.
All bands of radio waves are used for radio broadcasting.
The television circuit is almost identical to the radio broadcast circuit. The difference is that in the transmitter, vibrations are modulated not only by sound signals, but also by image signals. In a transmission telecamera, the image is reconstructed using an electron beam tube. The transmitted and received signals are synchronized in such a way that the motion of the electron beam in the television tube reproduces the motion of the beam of the transmitting television camera.
Currently, using electromagnetic waves, it is possible to transmit images of stationary and moving objects (phototelegraphy, television), control aircraft and ships (radio navigation), and accurately measure the distance under the Earth (radiogeodesy). With the help of radio antennas and radio telescopes, it became possible to radioprobe objects located in very distant points of space and receive the waves coming from them.
Review Questions:
-
What vibrations are called electromagnetic vibrations?
-
How is the vibration contour constructed?
-
What types of energy are involved in electromagnetic vibrations?
-
Write the formula for the period of oscillation in the oscillation circuit without active resistance.
-
What kind of electromagnetic oscillations are generated in an active resistive circuit?
-
What kind of vibrations are called forced electromagnetic vibrations?
-
What is an electromagnetic wave?
-
What is the scale of electromagnetic waves?
Test questions:
-
What contour is called an ideal vibration contour?
-
Write Thomson's formula.
-
How is the differential equation of damped electromagnetic oscillations expressed?
-
what is the expression called?
-
What is the resonance frequency?
-
What are the ideas behind Maxwell's electromagnetic wave theory?
-
What is the wavelength of an electromagnetic wave and how is it related to the period of oscillation and the speed of propagation of the wave?
-
Let's write the formula that expresses the speed of electromagnetic wave propagation in the medium.
-
What is the meaning of the Umov-Poynting vector and what is its formula?